According to the Oxford Dictionary a musician is "A person who plays a musical instrument, especially as a profession, or is musically talented."

According to the same dictionary a singer is "A person who sings, especially professionally."

To answer the question posed in the OP

"I am a singer with no musical training. I can sing in tune but play no instrument. Can I class myself as a 'musician'?

I would say no you cannot classify yourself as a "musician", by your own admission you say that you have "no musical training". There are of course varying degrees and I would think requirements. Personally I do not think that anyone who cannot read music can truly call themselves a musician.

By your description you are a singer that can sing in tune and hold a tune which means that you have been born with a wonderful and natural gift. Count your blessings there are many "professional performers" who cannot sing to save themselves.