The second statement is true.Let's take an example:
$\begin{bmatrix} 1&5 \\ 3 &6 \end{bmatrix}$ is a square matrix and its transpose is $\begin{bmatrix} 1 &3 \\5 & 6 \end{bmatrix}$ which is not equal to original matrix.
And transpose of any diagonal matrix, let's say $\begin{bmatrix} 1 &0 &0 \\ 0& 5 &0 \\ 0&0 &8 \end{bmatrix}$ and its transpose is $\begin{bmatrix} 1 &0 &0 \\ 0& 5 &0 \\ 0&0 &8 \end{bmatrix}$