We work over $\mathbb{C}$. Let $A\in M_n(\mathbb{C})$. (so our vector space is finite dimensional.) We know that
$A$ is unitary iff $AA^*=I=A^*A$ where $^*$ is conjugate transpose.
$A$ is normal iff $AA^*=A^*A$.
In particular unitary matrices are normal.
$A$ is unitarily diagonalizable iff there is a unitary matrix $P$ such that $PAP^{-1}$ is a diagonal matrix.
Question 1: Is unitarily diagonalizable the same as orthonormally diagonalizable in some books? I suppose unitary matrices are just orthonormal matrices with respect to the inner product $\langle v,w\rangle=v^*w$?
I am also quite confused between the unitary matrix and a matrix that is unitarily diagonalizable. The spectral theorem states
$A$ is normal iff $A$ is unitarily diagonalizable.
I think a corollary of spectral theorem is
$A$ is unitary iff $A$ is unitarily diagonalizable and all eigenvalues having absolute value $1$.
Combining these results it seems we get
$A$ is unitary iff $A$ is normal with all eigenvalues having absolute value $1$.
That is the distinction between unitary and unitarily diagonalizable matrices.
Question 2: Are the above statements correct? If not, could you provide an example.