Waistcoat vs vest

There are differences from one side of the Atlantic to the other as well. I'm going to let a BrE speaker answer, because I think that's what you will want.

We don't use the word "waistcoat" - we think of if as something that the White Rabbit would wear in Alice in Wonderland.

A vest is one of two things: either it's part of a man's three-piece suit: pants, jacket, and vest (note that our British friends wouldn't use the term "pants" either, I don't believe). It's the part that goes over the shirt, under the jacket. It doesn't have any sleeves.  It's difficult to see the vest, but you can see where the jacket ends; the vest is under it.

A vest can also be a knit sweater without sleeves.

Correct, we would not use 'pants' in the above.

In British English the term 'pants' = underpants worn by males.

You can also call something or someone 'pants' to mean they or it are rubbish/no good.

e.g. The England team are pants at scoring goals.