Look at your instrument pannel.
It has an Altimeter.
When you arrive in the cockpit and it is showing some random indication it is an Altimeter.
When you set QNH for departure, it is an Altimeter
When you set 1013 for enroute climb cruise and descent it is an Altimeter
When you set QFE for landing, it is an Altimeter.
Does your friend get confused and think that is should be called an Altimeter, a Flightlevelometer, and a Heighmeter?
The first uses of the radio altimeter was by the Navigator on long over-ocean crossings to work out "D" and over the ocean it most certainly measures Altitude.
Later uses for landing and (E)GPWS do not require us to call it something different.
RADio ALTimeter