The answer is going to be negative regardless of the names, so this geometric proof won't work.

3^2-2^2 =!= 2^2-3^2.

(You can exchange a and b in, say a^2+b^2, because 2^2+3^2=3^2+2^2)

This is not what I meant. What is being proved is: a^2-b^2 - (a+b)(a-b) = 0. If you swap a and b you end up with a sign switch on the lhs which is inconsequential.

That is not what the proof proves. The proof proves the equivalence how it was originally stated, and assumes for that b<a.

Your rewriting is of course true for all a,b and might be used in an algebraic proof. But this transformation is not at all shown in the geometric proof.

Did you think that I meant you can switch them on one side of the equation but not the other?

That's not what anyone is saying.

No, of course not.

But that's literally what you just did in your example.

I did not show the right side at all, so I am not sure how you can make that statement.

The point is that a+b is symmetric in a <-> b and a-b is anti-symmetric. Both left and right side are anti-symmetric.