A pinching has the form $M \mapsto T * M$, where $*$ is the entrywise product and $T$ is a $0/1$-matrix. I have the impression that (1)-(3) are all equivalent to
(4) T is positive,
and that it can be proved via the following: if a $0/1$-matrix $T$, with $1$ on the diagonal, avoids the pattern
$\left( \begin{array}{ccc} 1 & 1 & 0 \\ 1 & 1 & 1 \\ 0 & 1 & 1 \end{array} \right)$, right)$, then $T$ must be block-diagonal.
Either (1) or (2) imply that $T$ avoids this pattern: for (1) via your example, for (2) because it implies (4) (apply the pitching to the matrix with all entries equal to 1).
