1 Timothy 4:7-10
But refuse profane and old wives' fables, and exercise thyself rather unto godliness. For bodily exercise profiteth little: but godliness is profitable unto all things, having promise of the life that now is, and of that which is to come. This is a faithful saying and worthy of all acceptation. For therefore we both labour and suffer reproach, because we trust in the living God, who is the Saviour of all men, specially of those that believe.
Notice that there are a set of parallels that build up to verse 10.
1) refuse profane and old wives' fables || exercise thyself rather unto godliness.
2) For bodily exercise profiteth little || but godliness is profitable unto all things,
3) [Implied that bodily exercise has a promise in this life] || [godliness has] promise of the life that now is, and of that which is to come.
This is a faithful saying and worthy of all acceptation. For therefore we both labour and suffer reproach, because we trust in the living God,
4) who is the Saviour of all men || specially of those that believe
The point then from that we get in the fourth parallel, utilizing the contextual indications, is that the living God (the God of Life) is both the Savior with respect to preserving the physical lives of his creatures but with respect to believers their Savior in the special sense of giving them eternal life. All life comes from God, both our physical life and eternal life as well.
This is confirmed from later in the text when Paul tells Timothy:
1 Timothy 6:17-19
Charge them that are rich in this world, that they be not highminded, nor trust in uncertain riches, but in the living God, who giveth us richly all things to enjoy; that they do good, that they be rich in good works, ready to distribute, willing to communicate; laying up in store for themselves a good foundation against the time to come, that they may lay hold on eternal life.
Notice that the living God gives us both the riches of this life, but also of the life to come. Notice indeed the similarity between being more concerned about doing good and laying up for the life to come, rather than on this life.
I realize that's a short exegesis, but I think that it commends itself over any exegesis that argues that "Saviour of all men" should be understood as being a savior in a potential way. That sense leaves the phrase like a fish out of water, with no ties to the context.