I don’t remember making this assumption in the OP, but it’s this kind of answer which is most useful: not the statistical number of mathematically possible configurations, but the number of actual combinations one can make with an actual cube.
Well, most of the answers (including mine, CJJ*'s, and CalMeacham’s) involve actual cubes. I’m not sure quite what the “statistical number of mathematically possible configurations” is supposed to mean; it sounds like you think we’re making estimates or bounds. But my answer of “96” is not a bound, it’s the actual answer (assuming CJJ* and I have calculated correctly). I could list them all, and manipulate a real cube into any of them without any disassembly.
The difficulty is that the number of actual combinations you can make that look the same on the three front faces depends on what you want the faces to look like. There are exactly 96 cube states that look solved on the front faces; these are due entirely to moving edges about (since, as garygnu also points out, the corner positions can all be deduced).
For some front-face appeances (like the three I described earlier) there are more than 96 cube states that look the same. For the first case, for example, there are six possible corner states and so 6*96=576 possible states for the cube.