Cross: "Christ’s being eternally begotten of the Father refers to the procession of the Son from the Father. That is, the Logos eternally proceeds from the Father."
Bryan Cross is a member of the Roman communion and putatively some sort of teacher of his church's doctrine via his website, Called to Communion.
Roman theology, however, actually makes a distinction between begetting and procession. The Spirit proceeds from the Father (and the Son) but the Son is begotten of the Father. In fact, the Council of Florence defined things this way:
Whoever wills to be saved, before all things it is necessary that he holds the catholic faith. Unless a person keeps this faith whole and undefiled, without doubt he shall perish eternally. The catholic faith is this, that we worship one God in the Trinity, and the Trinity in unity, neither confounding the persons nor dividing the substance. For there is one person of the Father, another of the Son, and another of the holy Spirit. But the Godhead of the Father, of the Son and of the holy Spirit is one, the glory equal, and the majesty co-eternal. Such as the Father is, such is the Son, and such is the holy Spirit. The Father uncreated the Son uncreated and the holy Spirit uncreated. The Father infinite, the Son infinite and the holy Spirit infinite. The Father eternal, the Son eternal and the holy Spirit eternal. Yet they are not three eternals, but one eternal. As also they are not three uncreateds nor three infinites, but one uncreated and one infinite. Likewise the Father is almighty, the Son is almighty and the holy Spirit is almighty. Yet they are not three almighties, but one almighty. Likewise the Father is God, the Son is God and the holy Spirit is God. Yet they are not three gods, but one God. Likewise the Father is Lord, the Son is Lord and the holy Spirit is Lord. Yet they are not three lords, but one Lord. For just as we are compelled by the Christian truth to acknowledge each person by himself to be God and Lord, so we are forbidden by the catholic religion to say there are three gods or three lords. The Father is made by none, neither created nor begotten. The Son is from the Father alone; not made nor created, but begotten. The holy Spirit is from the Father and the Son; not made nor created nor begotten, but proceeding. So there is one Father, not three fathers; one Son, not three sons; one holy Spirit, not three holy spirits. And in this Trinity nothing is before or after, nothing is greater or less; but the whole three persons are co-eternal together and co-equal. So that in all things, as has been said above, the unity in Trinity and the Trinity in unity is to be worshipped. Whoever, therefore, wishes to be saved, let him think thus of the Trinity.The council goes on to reemphasize this: "The Father alone from his substance begot the Son; the Son alone is begotten of the Father alone; the holy Spirit alone proceeds at once from the Father and the Son."
The implication of the Council of Florence's statement is that someone like Cross, who alleges that the Son proceeds from the Father ("the procession of the Son from the Father") might not be saved, because he doesn't think of the Trinity the way the Council of Florence did. To some extent, that's Rome arrogance with respect to the Filioque, as though they get to define the gospel so as to exclude the Greeks from it. But we can address Rome's arrogance another time.
On another tangent, while Jordan doesn't appear to commit the identical basic problem, Jordan does seem to confuse "begotten from all eternity" with "eternally begetting" (see Jordan's comments in the comment box), in other words he is confusing a fait accompli with an on-going action. Thus, Jordan makes bizarre statements like "The Son eternally becomes mature" and "The Spirit eternally causes the Son to mature," neither of which appears to have any legitimate basis in the Scriptures (or in Tradition, i.e. church history, for that matter).
Bryan Cross has written a follow-on in which he appeals to Thomas Aquinas, who describes the "generation" of the Son as a type of procession. The only problem with this, of course, is that Thomas died in 1274, and the Council of Florence was in the mid-1400s. Bryan Cross can't appeal to Thomas Aquinas in order to deny the immaculate conception against Ineffabilis Deus, and he can't appeal to Thomas in order to deny something that Florence said.
Nevertheless, this way of speaking on Bryan's part is not entirely without precedent in Roman theology post Florence. John Paul II used the term "procession" to refer to the eternal generation of the Son in a general audience on 20 November 1985. There, JP2 distinguishes between spiration and generation but describes both as "procession." So, perhaps I am being unduly harsh on Bryan in insisting that he maintain the distinctions set forth in Florence when even the second most recent pope doesn't keep them straight.
At least, shall we say, JP2 and Cross do not maintain the exclusive use of "procession" with reference to spiration that Florence did. For example, note in Session 6, the following explanation:
The Latins asserted that they say the holy Spirit proceeds from the Father and the Son not with the intention of excluding the Father from being the source and principle of all deity, that is of the Son and of the holy Spirit, nor to imply that the Son does not receive from the Father, because the holy Spirit proceeds from the Son, nor that they posit two principles or two spirations; but they assert that there is only one principle and a single spiration of the holy Spirit, as they have asserted hitherto.
To wrap up, I think my words "demonstrated his lack of familiarity" may be unduly harsh and unjustified, so I retract them in favor "demonstrated a departure from the dogmatically defined distinctions employed by the Council of Florence." After all, perhaps Bryan is more familiar with Aquinas' usage than with the subsequent dogmatic definition of Bryan's church, or perhaps Bryan is influenced by John Paul II's usage
One assumes that Florence, on the other hand was more influenced by Isidore of Seville's ancient distinctions (either directly or indirectly), for the second item posted above seems to be almost a verbatim quotation from his Etymologies (The Etymologies VII.iv.4; see also VII.iii.6-8).