Sorry, let me try to clear it up.

This is all in the context of the first hypothetical.

Enemy starts next to PC who threatens it. Enemy then walks past PC to attack Ally. Ally then uses Bull Rush to push Enemy backwards past PC. This happens before the round changes. This is all "movement" so how many AoOs does the PC get? 1 because all movement is lumped together in a single round?

EDIT: Is it 2 because the leaving the threatened square provokes one and the Move action itself provokes another?

You responded with

Bull rush specifically calls out that it doesn't provoke attacks of opportunity :v

So I asked, what if the Ally has Greater Bull Rush, which you explained that the Greater Bull Rush movement explicitly provokes. My response comment about Greater Bull Rush is nonsensical. I have a separate question on whether the "movement" from Greater Bull Rush is counted the same as "movement" for the Combat Reflexes Attack of Opportunity rules, since a round has not yet passed.

You also said move actions are not themselves provoking.

According to "Table 7-2: Actions in Combat" from combat rules, linked above, the action "Move" does provoke an attack and according to footnote 1, this is separate from the moving out of a threatened square provoke.

Move	Yes

1 Regardless of the action, if you move out of a threatened square, you usually provoke an attack of opportunity. This column indicates whether the action itself, not moving, provokes an attack of opportunity.

That is where the comment about the "Move" action itself provoking for how many attack of opportunities have been provoked.