Why shouldn't a DJ be considered a musician?

We usually hear that many people that aren't in the electronic scene say that a DJ isn't a musician, that he doesn't actually 'play' music. Some say that electronic music is not real music, but let's not focus on this now.
So, let's compare what a typical musician does and what a dj does.

Musician	DJ
Performs both live and in studio	Performs both live and in studio
Mixes notes to create a melody	Mixes tracks to create a DJ-set
Can rely on improvisation during live performances	Can rely on improvisation during live performances
Performs with a musical instrument, using his technical ability	Performs with a console, using his technical ability

So by only switching words we get more or less the same thing. The only one that can be difficult to explain is the second point. Let's think about it a bit metaphysically.
So let's imagine a whole song is a single note. And let's say we have seven songs called A, B, C, D, E, F, G and we arrange them like this:
A-F-B-E-G-D-C
Next we take the seven notes (in the anglosaxon notation), that is A, B, C, D, E, F, G and we arrange them like this:
A-F-B-E-G-D-C
Do you see any difference? I don't.
So it's just a metaphysical argument? I think so.