I distinguish them this way:
- if the barrel screws into it, it is a receiver; i.e. "receives" the barrel
- if integral with the barrel, it is a barrel extension.
Thus the Luger has a receiver, and the Mauser broom handle as made has a barrel extension.
The AR platform has both an upper receiver and a lower receiver, just to keep things interesting.
And a frame can also be a receiver; but a Luger frame is not a receiver, the Mauser frame is also a receiver; but, in most revolvers the frame is the receiver. Confusing enough for me!