What do you mean by $CZ_{1,2,3}$? You should specify which ones are control and which ones are being acted on.
It is a (useful such as for graph states) coincidence that for $CZ_{1,2}$ you get the same result by interpreting either as the control and the other as being acted on.
$$
\begin{pmatrix}
1&0&0&0\\
0&1&0&0\\
0&0&1&0\\
0&0&0&-1\\
\end{pmatrix}
$$
is invariant under the action of permuting the tensor factors. That is by conjugating by the following matrix:
$$
\begin{pmatrix}
1&0&0&0\\
0&0&1&0\\
0&1&0&0\\
0&0&0&1\\
\end{pmatrix}
$$
but that is not generally true.
$CU_{1,2}$ with the $1$ qubit acting as control and $2$ being acted on by the unitary 2 by 2 matrix $U$ in that case. In total this would be the 4 by 4 unitary matrix (again in usual basis)
$$
\begin{pmatrix}
1&0&0&0\\
0&1&0&0\\
0&0&u_{00}&u_{01}\\
0&0&u_{10}&u_{11}\\
\end{pmatrix}
$$
You can easily see that in this general case, it matters which is the control and which is acted on. It is not invariant under exchange of factors.
So the first thing you wrote is the result of interpreting the first as the control and acting on the third. The second thing you wrote means you have two controls. Both 1 and 2 are controls in that case. Again acting on the third.