I should start by saying that everything I'm about to say is just my thought process and opinion on this question, the crowd might be divided on this but I could be completely off base, maybe I'm wrong (it wouldn't be the first time).
For years I was a singer/vocalist in different bands and I really don't know how I feel about this question, there are so many different factors to consider and to draw definite lines is near impossible to me.
Personally, I would define myself as a musician, even though I may not have been playing an instrument in these bands, I could/can play, and I'm more than capable of writing a half decent song (although that could be subjective). So of course, I can confidently say I am a musician.
Within the setting of a band, I think even if you are instrument-less, you can still be a musician, after all, mosts bands write music together and as a singer, you're always listening to sounds, making writing notes, creating melodies and moulding something from nothing alongside and equal to other members. So it would stand to reason that just because you're writing music without playing an instrument doesn't mean you aren't making music as a musician.
Then there is the question, what is a musical instrument? Well, to me it's anything that produces a sound that you can create music and/or songs with, computers fall into that category for me, and so I THINK my opinion is that even if you're programming and sequencing music, you're still a musician, surely? Maybe not in the 'traditional' sense, but then times do change, and so things like this have to change too I guess.
So far I'm pretty sure that if you are a singer in a band, someone who makes music electronically, or just a singer that can and does contribute to the musical aspect of songwriting in any environment then I'm pretty sure I'd class you as a musician, but let's say you are a singer and just that. You don't write songs or melodies, you just perform the song as it's presented to you and don't hold any ability to play a musical instrument, wait a minute... Isn't your voice a 'musical instrument? For the sake of this point, let's say it's not.
There is a difference between a performer and a musician, but they are both 'artists' I wouldn't call someone who takes a song, as it's given to them, performing it as it's already been written - without changing any of the musicality in the track - a musician. I would, however, call them a performer and an artist. My view of the music world is that two industries have accidentally occupied the same space. Those industries are the musician's industry and the musical performer's industry, together they're just called 'the music industry' but they are vastly different worlds. One appeals to the art of making music and the other appeals to the art of performance, they occupy the same space and so to the average listener they are taken away as the same thing. That's just my view and how I see music on a broader scale.
So, I guess to summarize, yes I think a singer is a musician given that the circumstances are that there is musical input when creating the music/song. Equally, you can be a singer but not a musician if you are simply performing what has already been written. However, in both scenarios, you're still an artist just two different types of artists occupying the same space. I don't think the two are mutually exclusive.
Like I said in the start I could be wrong. What do you think the answer is? is there even one out there?