To be honest the second image with the tilted c² is enough. From that picture alone you can figure the outer square has the same area as the inner square + 4 times the triangle area:

(a+b)² = c² + 4(ab/2)

a² + b² + 2ab = c² + 2ab

a² + b² = c²

Yours does use some elementary algebra[1], which wasn't used in the Chinese geometric proof. I wonder if ancient Chinese mathematicians could simplify (a+b)^2 symbolically, or even did symbolic algebra this way?

(When I referred to "ancient Chinese visual proof" I wasn't bullshitting, but I didn't find a source with the exact 2-part picture, though this makes same claim using same pictures: http://www.researchhistory.org/2012/10/24/earliest-evidence-... )

I'm sure they knew that the area of triangles, which you also use, is (ab/2) - but the purely visual proof needs nothing more than the knowledge that the area of a square is the square of the length of its sides. (And I guess some obvious facts like that no matter how you divide an area the sum of the areas of its parts will be same - the reason I mention tiny slivers is in some fake geometric proofs this intuitive knowledge is abused.

For example, see this excellent description:

https://en.m.wikipedia.org/wiki/Missing_square_puzzle

Before you open the solution, you can look at the trick for as long as you want, you won't figure it out.)

[1] https://en.m.wikipedia.org/wiki/FOIL_method