What are the differences between terminal types supported by iTerm2?
Short bet, go for xterm-256color

The setting in iTerm2 affects two things:
1. How the TERM environment variable is initially set. Your login scripts are changing this if it gets set to screen (or you're using screen or tmux, which always set it to screen)
2. The "ansi" terminal automatically scrolls when the cursor is on the bottom right of the screen. You probably don't want this.

As for how the TERM var is interpreted by apps, that's complicated. Most people want xterm-256color unless they're sshing to a host that doesn't support it. In order of capability and support, I'd order them:
xterm-256color
xterm-new
xterm
vt100

I probably wouldn't use the others unless I had a really good reason to (they're carried over from the original iTerm code and may or may not work well).

Anyway, figure out why your TERM var is getting changed to screen and that's probably the cause of your trouble.