Occasionally people will complain that the term "semi-pelagianism" gets thrown around too freely. As an antidote, here are some comments from noted historian Philip Schaff:
Semi-Pelagianism is a somewhat vague and indefinite attempt at reconciliation, hovering midway between the sharply marked systems of Pelagius and Augustine, taking off the edge of each, and inclining now to the one, now to the other. The name was introduced during the scholastic age, but the system of doctrine, in all essential points, was formed in Southern France in the fifth century, during the latter years of Augustine’s life and soon after his death. It proceeded from the combined influence of the pre-Augustinian synergism and monastic legalism. Its leading idea is, that divine grace and the human will jointly accomplish the work of conversion and sanctification, and that ordinarily man must take the first step. It rejects the Pelagian doctrine of the moral roundness of man, but rejects also the Augustinian doctrine of the entire corruption and bondage of the natural man, and substitutes the idea of a diseased or crippled state of the voluntary power. It disowns the Pelagian conception of grace as a mere external auxiliary; but also, quite as decidedly, the Augustinian doctrines of the sovereignty, irresistibleness, and limitation of grace; and affirms the necessity and the internal operation of grace with and through human agency, a general atonement through Christ, and a predestination to salvation conditioned by the foreknowledge of faith. The union of the Pelagian and Augustinian elements thus attempted is not, however, an inward organic coalescence, but rather a mechanical and arbitrary combination, which really satisfies neither the one interest nor the other, but commonly leans to the Pelagian side.
For this reason it admirably suited the legalistic and ascetic piety of the middle age, and indeed always remained within the pale of the Catholic church, and never produced a separate sect.
We glance now at the main features of the origin and progress of this school.
The Pelagian system had been vanquished by Augustine, and rejected and condemned as heresy by the church. This result, however, did not in itself necessarily imply the complete approval of the Augustinian system. Many, even opponents of Pelagius, recoiled from a position so wide of the older fathers as Augustine’s doctrines of the bondage of man and the absolute election of grace, and preferred a middle ground.
First the monks of the convent of Adrumetum in North Africa differed among themselves over the doctrine of predestination; some perverting it to carnal security, others plunging from it into anguish and desperation, and yet others feeling compelled to lay more stress than Augustine upon human freedom and responsibility. Augustine endeavored to allay the scruples of these monks by his two treatises, De gratia et libero arbitrio, and De correptione et gratia. The abbot Valentinus answered these in the name of the monks in a reverent and submissive tone.
But simultaneously a more dangerous opposition to the doctrine of predestination arose in Southern Gaul, in the form of a regular theological school within the Catholic church. The members of this school were first called “remnants of the Pelagians,” but commonly Massilians, from Massilia (Marseilles), their chief centre, and afterwards Semi-Pelagians. Augustine received an account of this from two learned and pious lay friends, Prosper, and Hilarius, who begged that he himself would take the pen against it. This was the occasion of his two works, De praedestinatione sanctorum, and De dono perseverentiae, with which he worthily closed his labors as an author. He deals with these disputants more gently than with the Pelagians, and addresses them as brethren. After his death (430) the discussion was continued principally in Gaul; for then North Africa was disquieted by the victorious invasion of the Vandals, which for several decades shut it out from the circle of theological and ecclesiastical activity.
At the head of the Semi-Pelagian party stood John Cassian, the founder and abbot of the monastery at Massilia, a man of thorough cultivation, rich experience, and unquestioned orthodoxy.He was a grateful disciple of Chrysostom, who ordained him deacon, and apparently also presbyter. His Greek training and his predilection for monasticism were a favorable soil for his Semi-Pelagian theory. He labored awhile in Rome with Pelagius, and afterwards in Southern France, in the cause of monastic piety, which he efficiently promoted by exhortation and example. Monasticism sought in cloistered retreats a protection against the allurements of sin, the desolating incursions of the barbarians, and the wretchedness of an age of tumult and confusion. But the enthusiasm for the monastic life tended strongly to over-value external acts and ascetic discipline, and resisted the free evangelical bent of the Augustinian theology. Cassian wrote twelve books De coenobiorum institutis, in which be first describes the outward life of the monks, and then their inward conflicts and victories over the eight capital vices: intemperance, unchastity, avarice, anger, sadness, dulness, ambition, and pride. More important are his fourteen Collationes Patrum, conversations which Cassian and his friend Germanus had had with the most experienced ascetics in Egypt, during a seven years’ sojourn there.
In this work, especially in the thirteenth Colloquy, he rejects decidedly the errors of Pelagius, and affirms the universal sinfulness of men, the introduction of it by the fall of Adam, and the necessity, of divine grace to every individual act. But, with evident reference to Augustine, though without naming him, he combats the doctrines of election and of the irresistible and particular operation of grace, which were in conflict with the church tradition, especially, with the Oriental theology, and with his own earnest ascetic legalism.
In opposition to both systems he taught that the divine image and human freedom were not annihilated, but only weakened, by the fall; in other words, that man is sick, but not dead, that he cannot indeed help himself, but that he can desire the help of a physician, and either accept or refuse it when offered, and that he must cooperate with the grace of God in his salvation. The question, which of the two factors has the initiative, he answers, altogether empirically, to this effect: that sometimes, and indeed usually, the human will, as in the cases of the Prodigal Son, Zacchaeus, the Penitent Thief, and Cornelius, determines itself to conversion; sometimes grace anticipates it, and, as with Matthew and Paul, draws the resisting will—yet, even in this case, without constraint—to God. Here, therefore, the gratia praeveniens is manifestly overlooked.
These are essentially Semi-Pelagian principles, though capable of various modifications and applications. The church, even the Roman church, has rightly emphasized the necessity of prevenient grace, but has not impeached Cassian, who is properly the father of the Semi-Pelagian theory. Leo the Great even commissioned him to write a work against Nestorianism, in which he found an excellent opportunity to establish his orthodoxy, and to clear himself of all connection with the kindred heresies of Pelagianism and Nestorianism, which were condemned together at Ephesus in 431. He died after 432, at an advanced age, and though not formally canonized, is honored as a saint by some dioceses. His works are very extensively read for practical edification.
Against the thirteenth Colloquy of Cassian, Prosper Aquitanus, an Augustinian divine and poet, who, probably on account of the desolations of the Vandals, had left his native Aquitania for the South of Gaul, and found comfort and repose in the doctrines of election amid the wars of his age, wrote a book upon grace and freedom, about 432, in which he criticises twelve propositions of Cassian, and declares them all heretical, except the first. He also composed a long poem in defence of Augustine and his system, and refuted the “Gallic slanders and Vincentian imputations,” which placed the doctrine of predestination in the most odious light.
But the Semi-Pelagian doctrine was the more popular, and made great progress in France. Its principal advocates after Cassian are the following: the presbyter-monk Vicentius of Lerinum, author of the Commonitorium, in which he developed the true catholic test of doctrine, the threefold consensus, in covert antagonism to the novel doctrines of Augustinianism (about 434); Faustus, bishop of Rhegium (Riez), who at the council of Arles (475) refuted the hyper-Augustinian presbyter Lucidus, and was commissioned by the council to write a work upon the grace of God and human freedom; ennadius, presbyter at Marseilles (died after 495), who continued the biographical work of Jerome, De viris illustribus, down to 495, and attributed Augustine’s doctrine of predestination to his itch for writing; Arnobius the younger; and the much discussed anonymous tract Praedestinatus (about 460), which, by gross exaggeration, and by an unwarranted imputation of logical results which Augustine had expressly forestalled, placed the doctrine of predestination in an odious light, and then refuted it.
The author of the Praedestinatus says, that a treatise had fallen into his hands, which fraudulently bore upon its face the name of the Orthodox teacher Augustine, in order to smuggle in, under a Catholic name, a blasphemous dogma, pernicious to the faith. On this account he had undertaken to transcribe and to refute this work. The treatise itself consists of three books; the first, following Augustine’s book, De haeresibus, gives a description of ninety heresies from Simon Magus down to the time of the author, and brings up, as the last of them, the doctrine of a double predestination, as a doctrine which makes God the author of evil, and renders all the moral endeavors of men fruitless; the second book is the pseudo-Augustinian treatise upon this ninetieth heresy, but is apparently merely a Semi-Pelagian caricature by the same author; the third book contains the refutation of the thus travestied pseudo-Augustinian doctrine of predestination, employing the usual Semi-Pelagian arguments.
A counterpart to this treatise is found in the also anonymous work, De vocatione omnium gentium, which endeavors to commend Augustinianism by mitigation, in the same degree that the Praedestinatus endeavors to stultify it by exaggeration. It has been ascribed to pope Leo I. († 461), of whom it would not be unworthy; but it cannot be supposed that the work of so distinguished a man could have remained anonymous. The author avoids even the term praedestinatio, and teaches expressly, that Christ died for all men and would have all to be saved; thus rejecting the Augustinian particularism. But, on the other hand, he also rejects the Semi-Pelagian principles, and asserts the utter inability of the natural man to do good. He unhesitatingly sets grace above the human will, and represents the whole life of faith, from beginning to end, as a work of unmerited grace. He develops the three thoughts, that God desires the salvation of all men; that no one is saved by his own merits, but by grace; and that the human understanding cannot fathom the depths of divine wisdom. We must trust in the righteousness of God. Every one of the damned suffers only the righteous punishment of his sins; while no saint can boast himself in his merits, since it is only of pure grace that he is saved. But how is it with the great multitude of infants that die every year without baptism, and without opportunity of coming to the knowledge of salvation? The author feels this difficulty, without, however, being able to solve it. He calls to his help the representative character of parents, and dilutes the Augustinian doctrine of original sin to the negative conception of a mere defect of good, which, of course, also reduces the idea of hereditary guilt and the damnation of unbaptized children. He distinguishes between a general grace which comes to man through the external revelation in nature, law, and gospel, and a special grace, which effects conversion and regeneration by an inward impartation of saving power, and which is only bestowed on those that are saved.
Semi-Pelagianism prevailed in Gaul for several decades. Under the lead of Faustus of Rhegium it gained the victory in two synods, at Arles in 472 and at Lyons in 475, where Augustine’s doctrine of predestination was condemned, though without mention of his name.
(source) (emphasis added)
That provides a very lengthy explanation, but you will find that at other places Schaff simply uses Semi-Pelagian as a synonym for synergistic. For example, "in reference to the freedom of the will and predestination he adopted synergistic or Semi-Pelagian views," (source) (emphasis added) or "The position of the Greek church upon this question is only negative; she has in name condemned Pelagianism, but has never received the positive doctrines of Augustine. She continued to teach synergistic or Semi-Pelagian views, without, however, entering into a deeper investigation of the relation of human freedom to divine grace." (source) (emphasis added) "And yet we must say that the Reformers, following the lead of the great saint of Hippo, went to a one-sided extreme. Melanchthon felt this, and proposed the system of synergism, which is akin to the semi-Pelagian and Arminian theories. Oecolampadius kept within the limits of Christian experience and expressed it in the sound sentence, "Salus nostra ex Deo, perditio nostra ex nobis." We must always keep in mind both the divine and the human, the speculative and the practical aspects of this problem of ages; in other words, we must combine divine sovereignty and human responsibility as complemental truths. There is a moral as well as an intellectual logic,—a logic of the heart and conscience as well as a logic of the head. The former must keep the latter in check and save it from running into supralapsarianism and at last into fatalism and pantheism, which is just as bad as Pelagianism." (source) (emphasis added)