The OED combines these two definitions:
1b. In effect, although not according to strict definition; to all intents and purposes; as good as; practically. Later frequently in weakened use: nearly, almost.
Personally, I think that is a better way to understand the term, and how it is regarded in English. These are not two separate meanings with separate etymologies, but the "almost entirely" meaning is simply a weakening of the "for all practical purposes" meaning. It's not hard to see that often when A is "effectively" or "for all practical purposes" B, it is so because A is "nearly" B in some sense. There will be some exceptions (and this is arguably one of them, see below) but there is a lot of overlap between the two meanings.*
This leaves open the question of which is being used in your example, although it illustrates why the difference between the two meanings is regarded as being razor-thin.
With that in mind, three answers:
1) This seems like a classic case of a distinction without a difference. Given that the statement is almost certainly not meant to be taken in a completely literal manner, nailing down which shade of the definition was in the speaker's mind doesn't seem to add much to the understanding of the meaning.
2) The "for all practical purposes" seems like the definition that would be slightly more useful in making the speaker's point. Something could be infinite "for all practical purposes" or "as good as" infinite, without being demonstrated to be literally infinite. It is possible that the context might indicate that the speaker was actually making a claim about the literal numerical vastness of the unsupported things he or she believes, but that seems to be less important than a more general point about his or her credulity/open-mindedness.
3) Strictly speaking, it's impossible for something to be "nearly infinite" in a literal sense; it either is or isn't. Logically, the speaker must mean "infinite for all intents and purposes".
(I realize I've elided infinite and endless through this answer, even though the quote technically modifies endless with virtually, but I don't think it changes any of the points.)
*I would argue, based entirely off of my own intuition, and I have no references to back it up, that the two meanings are even closer than this implies: that even when we use virtually to simply mean nearly or almost that we only use it when the difference doesn't much matter or when we are minimizing the difference. If someone collapses short of the finish line, we could say that he nearly finished the race, but not that he virtually finished, because the whole point of a race is to cross the finish line, unless we were intentionally trying to minimize the fact that he didn't finish.