writes:

I'm not from the US, but just wondering if you depart ifr from an
airport and tower tells you to contact departure, I know that you
check in with callsign, altitude through, and cleared altitude.

But I was wondering if the AIM recommends anywhere also to include the
departure runway and the departure airport too.

They already know that from the handoff. They know your altitude, too, but
you give it so that they can verify that the encoded altitude from the
transponder matches the altitude you see on your instruments.