This may be just a reiteration of what Ken and David have already said, but here are my two cents worth:
For the record and for your friend, HDTV in theory can be analog or digital (it is not necessarily a subset of DTV). It is exactly what it says it is--HIGH DEFINITION television (i.e. 720p or 1080i lines of resolution, surround sound, and all that jazz). Nothing less, nothing more. One can get HIGH DEFINITION television over an analog signal, it would just take a lot more bandwidth. I am told that Japan in fact does broadcast HIGH DEFINITION television using analog signals.
The digital question has nothing to do with HIGH DEFINITION. Is is merely the streamlined method by which a signal gets from a production source to your television. Any signal can be broadcast in a traditional analog format (where the information is part of the moduluation of the radiowave) or in a digital format (where the information is encoded as a series of zeros and ones and the radiowave then carries the zeros and ones to the receiver). The primary advantages to digital signals are less bandwidth and less degradation of signal quality (thus the often heard statement, "crystal clear digital sound).
I BELIEVE the answer to your friend's question is WHEN and HOW MUCH of a signal is digital determines whether it is digital television or can be considered DTV.
Satellite broadcasters take a standard analog signal and digitize it before they uplink to the satellite. When the signal gets to your reciever, the receiver converts the digital information back into an analog format. Thus, they have broadcast a digital signal and can claim some level of digial television.
I BELIEVE that DTV implies the source signal itself is digital and is kept digital until it hits your receiver, which will then convert it to analog before it travels over the wires to your speakers and video source. But that is just my interpretation of DTV. THUS, DTV would have an even better quality of signal than digital television that is just digital in the last (although significant) stage of the broadcast stream.
Hope this helps.