Non-integral powers of a matrix

As @tom pointed out in a comment, the power of a matrix can be defined in terms of logarithm of a matrix and matrix exponential, using

$$A^p:=\exp\left(p\ln A\right)$$

Using the principal logarithm (this is the name for that choice described in the question without giving a name), the above even yields unique results.

The matrix exponential is defined for every matrix, the matrix logarithm only for invertible matrices. The case of singular matrices mentioned in the question is therefore not covered by this definition.

You are correct that your proposed definition for rational exponents can run into issues of uniqueness. Consider just the problem of trying to find the square root of a matrix. If $I$ is the 2x2 identity, then any matrix of the form

\begin{pmatrix} \pm1 & a \\ 0 & \mp1 \\ \end{pmatrix}

satisfies $A^2=I$. Now, there is a case where you can define a unique square root. In particular, your matrix must be positive definite [1]

For a more general discussion, see this