Lets keep it simple. The way I see it is.

A 'session' is a group of people getting together to play pedominately tunes (unless it's advertised 'mixed' or 'singing').
A 'singaround' is just that, a group of people taking turns to sing (or listening).
An 'open mike' does what it say's on the tin. They have a P.A. and it's basicaly a folk kareoke.
A 'singers night' usually refers to a folk club when they don't have a booked guest on, & it's a chance for both singers & instumentalists to do their stuff.

Simple init?
Cheers
Les