In the rules for NG+ it says that engage skip is allowed. Is that skipping dialogue before some missions, or is it something else altogether?

This refers to the "3, 2, 1, ENGAGE" at the start of every mission.

It turns out that restarting a mission from the pause menu or game over screen will skip the countdown and show "ENGAGE" right away. So if you pause and restart before the countdown even begins, you essentially skip it altogether.