The battery-charged cordless for the house I've always called a remote phone

Interesting one, Juan..

I'd never specify, maybe because we're almost always using the cordless phones (one generally upstairs and one downstairs) and hardly ever using the fixed ones. You get used to the freedom. I'd specify picking up the "fixed phone" sometimes, but not the cordless.

As is now obvious, though, if I did specify I'd say "cordless".

Oh, and you'd hope a cordless phone wasn't "remote" when somebody called.

Confusion between phones and remote controls.. hmmm, surprisingly rare actually. Maybe because the cordless phones we have are noticeably more narrow (side to side) and also more thick than our remote controls. They also have an LCD display, so they're more like mobies really.
They must be highly distinctive, as I'd definitely get them confused if there were the slightest opportunity to do so.