modal auxiliary 'could'

[beccagee]

1. She couldn't have seen him yesterday. Impossibility (past)
He's on holiday in Malta.

2. I couldn't open the door. Inability

3. Could you say that again? Request

4. We could either go to the pub or we could get a take-away. Possibility

Are there specific phrases to name a communicative function?

They aren't set in stone as absolute terms, but generally people tend to use the same words or similar ones.

[Casiopea]

Welcome

I would agree with your assumption.

. . . couldn't have . . . . (impossibility)
. . . couldn't. . . . (inability)
Could you . . . ? (request)
. . . could either . . . or I could . . . . (possibility)