As I see it, your (<=) proof is missing a key element: you have shown that if THERE IS a solution to Ax = b, it is unique. What you still need to do is show that some (particular) solution x0 EXISTS.
And here we have a problem: b may not be in the image space (column space) of A. To see what I mean, let m = 3, n = 2, and consider the matrix:
Then rank(A) = 2, but there is NO SOLUTION to Ax = b, since A(x,y,z)T = (x,y,0)T.
So what you are trying to "prove" IS NOT TRUE.
What IS true, is the following:
A linear system of m equations in n unknowns has AT MOST one solution if and only if rank(A) = n.