Though observant of you with the touch-tone and rotary phone, the phone on the actual episode was indeed a touch-tone phone. The Partridges were among the first television families to boast such a technology. They even had a touch-tone phone in their garage!
During the Partridge Family, Danny picks up the telephone. It is a push button phone. When the Keith and Lauriebelle, episode originally aired, the phone was probably rotary dial.
At the end of the Different Strokes segment when Lurlene is talking to Arnold and Willis, Kimberly can be see behind the front door (which is made of glass) waiting for her cue to make her entrance.