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Ugh! I hate him so much!

because he apparently doesn't know you're supposed to button the top button on your suit jacket (or the top two buttons, if it's a three-button jacket). Or maybe he doesn't so you can see the silken glory of his extra-long tie.

cover his butt and gut.

If you look at the clip in totality, you can see him coming up from behind in the crowd
He literally pushes people out of the way so he can be at the front of the crowd.

Once he is in front of the crowd, he stops in front of photographers. He looks
suddenly insecure and self-conscious. Like what do I do now? He thinks to pull his jacket
around him in a protective move an also to hide his weight.

And insecure.

It's his preening self-aggrandizing posture - classic male peacock.

Deeply wounded at the slightest affront and needs constant adoration to prop him up.
Constantly needs to devalue his imagined enemies.

My viewpoint is a clinical one that goes deeper than what is seen on the surface.

Some narcissists do have deep robust self-confidence, but Trump is not one of them.

Which shows up in his need to see only positive coverage about himself, his need to destroy anything Obama, from inaugural crowd size to DACA, his worship me cabinet meetings - although moreso in the fact that those who work for him feel the need to stage their interactions with him to avoid a two-year old style melt-down.

But this particular affectation isn't a pull-the-coat-around-me-for-protection. It is classic peacock preening. If it was based in what you suggest, rather than puffing himself up to appear bigger, his posture would have changed in the opposite direction (curling in, rather than puffing his chest out and jutting his jaw).

I think it is both grandiose as you suggest but also defensive.

The grandiosity, bravado and preening is a cover up for his fragile core.
I am just taking what you see down another layer.

He is like an onion, he has several layers.

And the look on his face is combative. So it's some sort of pre-fight thing maybe.

to make angry gestures.

I think he gets angry because he is in over his head, doesn't understand the rules,
and feels out of control.

in his case, I do think it's some kind of unconscious aggressive move, as you said.

hooking your thumbs in your suspenders and pulling them out.

On the one hand, it's "adjusting your clothing." OTOH, it's suggesting the puffing out of the chest like a rooster pushes out its chest feathers to look bigger.