You've found two infinite families and one sporadic solution.
There is another sporadic solution, and the 8 piece solution you've found is not, in fact a sporadic.
And yes, the infinities are uncountable.
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Oh, right - you can vary the shape of the diagonal lines, provided each "arm" has rotational symmetry about its mid-point.
OK, so we've got three infinite families and the trivial solution (only one piece - is that your sporadic solution?). I think that might be all: each piece can contain 4, 2, 1 or 1/2 of the original square's corners, since (lacuna) all pieces must contain the same number of corners and further subdividing the corners (into 1/3s, say) would mean some pieces don't touch the centre (another lacuna).
OK, that's now the set of solutions I've got. You've also gone some way to showing them to be complete.
More to do, though.
And now do it for an equilateral triangle.
OK - I've now seen a "solution" with 16 "pieces."
My head hurts.
Ah, yes. You can also rotate the 8-fold solution by 22.5 degrees for another solution.
Right. As I said, I'm weak in math.
You can't be that weak at what I call math, although your experiences with math education might be unhappy ones. You have found more solutions than I first found - that can't be bad.
Half my life is spent helping people discover that they're good at "proper math" even when they think they're bad at "school math."
Hmm. Don't think so.