Eigenvectors are NOT unique, for a variety of reasons. Change the sign, and an eigenvector is still an eigenvector for the same eigenvalue. In fact, multiply by any constant, and an eigenvector is still that. Different tools can sometimes choose different normalizations.

How do you know if eigenvalues are distinct?

"Distinct" numbers just means different numbers. If a and b are eigen values of operator T and then they are "distinct" eigenvalues. If they happen to be 0 and 1, then, since they are different, they are "distinct".

Can you have different eigenvectors?

If a matrix has more than one eigenvector the associated eigenvalues can be different for the different eigenvectors. ... Geometrically, the action of a matrix on one of its eigenvectors causes the vector to stretch (or shrink) and/or reverse direction.