Fallout – Set settlement spawn point

I have seen this extensively during my multiple playthroughs and believe this to be a bug of some kind. As you've noted, visiting the settlement and looking at the statistics in workshop mode reveal the correct numbers and once you've seen these numbers they'll be correctly reported in the Pip Boy interface.

This isn't limited to specific settlements and as far as I have seen, not related to specific events either (it's not going to happen more frequently if, for example, that settlement has just been attacked).

I have not found a method of ensuring these numbers are correct without visiting the settlement.

Nick doesn't leave the same way you came in. Instead, within viewable distance of where Skinny Malone (and thus the entrance to that particular vault is) there is an alley with a set of steps leading up to the surface. Usually after dealing with Skinny Malone you would both head over to this passage and climb the ladder before Nick makes a comment about the sky and discusses whether to travel with you back to Diamond City or whether he'll meet you there.

Nick will probably be stood waiting at the bottom of this ladder, or at the top of the ladder, outside of the subway station, waiting to have a conversation with you about the sky.