Let me know if anything is wrong with my proofs.

The third proof is really special.

A couple of considerations about solutions by Runner.

About Proof 1: here I try to show that the formula, that has been demonstrated when all eigenvalues are different each other, cannot be falsified when some of them are equal. This using general considerations, not the Jordan canonical form.
The formula is really an identity, because:
1.the left hand side, after having developed the exponential and then the determinant, is a series each term of which is a polynomial function of the eigenvalues ;
2. the right hand side, after having developed the trace and then the exponential, is a series each term of which is a polynomial function of the eigenvalues ;
3. the two series, being equals for every set of eigenvalues (different each other), have the corresponding terms (i.e. terms with same powers of eigenvalues) identical polynomials.
Being an identity, it becomes an equality for every set eigenvalues, even when some of them are equal.

About Proof 2: it is not demonstrated that multiplicity of eigenvalues is preserved by exponentiation; so, if I'm not wrong, the proof is assuming that all eigenvalues are different each other.