I'm inventing terms and phrases a little bit here but what would the difference be by displaying vulnerability and invulnerability?

It's both about displaying personal and truthful things about yourself but in the latter scenario you aren't sensitive about it.

You've dealt with all your pains by yourself and now you can admit anything, without emotional risk.

Doing so, the person you reveal it to cannot hurt you by gaining this knowledge. You've prestaged your reactions. You have no lingering shame or pain when you are sincere.

And that's a show of strength really.

If you get some jabs in return it's not malicious, most often. That's just her asking "Okay, but please assure me you're not some walking wound, are you?",

and your queue is to bounce it off with humor.

The thing is when women say "show vulnerability", what they actually mean is "be truthful". They want to know who they're getting involved with and that's very understandable.

They certainly don't want the hurt, prickly feel or tears to go with or the absolute death-knell:

The feeling that she somehow has to absolve him from this.

Fair enough, there is a small number who actually do want to "mother their boyfriends", and that's another discussion but I would hope a man doesn't want that woman, and rather suggest he does some serious introspection and completely exorcise that need first.

Rule of thumb, shame and remorse are NOT allowed emotions. They are only allowed in small doses when you fuck up for someone in the present. If you forget a coffee date and turn up 20 minutes late, yeah a tad of that is appropriate, but also bounce back.

Though the fact that you were bullied in grade school. That you happen to have a few uninhibitedly childish hobbies. That you sincerely like something morons would call "gay" etc.

That all comfortably rolls off your tongue with a wry smile.

If understood this way, yes, men should display "vulnerability".