I know that's not rule book text, but what criteria would you use to differentiate a foul tip from a caught fly ball?

When could a hit ball that never went above Batters head, that ended up behind the batter, caught by F2 be a caught fly out instead of a foul tip?

I can't picture it.
So though it might not be the "rule" practically speaking it's accurate.

If it's a foul ball hit in front of the batter and is caught makes no difference the height, but I dont know how F2 could catch it if it didn't pop-up above the head.

I've seen this in FED, though. The next time those two teams met on the schedule and finished the prior game first even though one team was leading when play was halted in the sixth.

Yep, that was my impression.

I think you have it "backwards". It's not "this is a fly and everything else is a foul tip." It's "this is a foul tip and everything else is a fly."

So, if it's not sharp and direct (iow, if there's any arc, even if it doesn't go above the head) it's a fly, and an out if it's caught.