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Thanks, mate!

The при- in приподняться means "a little". Pure "подняться" means "to stand up". Now that we have the "при", it means "to stand up a little", not completely. My English sucks when it comes to such...

Well, it's not a nod and not a bow, it's like in between them. When you rise a bit off your seat, take off your hat, move your hand with the hat down and aside, make a deep nod or a shallow bow (idk)...

To "sit up" is like when you straighten while sitting, if I'm not mistaken. But your butt (well, sorry for the literalness) is still in contact with the surface of the seat.

While "Приподняться"...

Yep, in response to his polite gesture of taking off his beret they had not but to take their backs off their seats and greet him with a bow.

Раскланяться is way more ceremonial than just кивнуть...