Are singers musicians?

That depends on your definition of musician. Loosely, anyone who makes music, vocal, percussion, or instrumental, is a musician.

If you hold the term musician to a higher standard, then only someone who is musically literate and has developed both the specialized knowledge and skills of music can be called a musician. Certainly, there are literate, skilled, and knowledgeable vocal musicians (Annie Lennox for example), but the average pop singer is not. And by this definition, the musicianship of many pop/rock instrumentalists could also be denied.

We might get around the dilemma by admitting Yoko and Rotten do NOT make music, therefore, they are NOT musicians.

Vocal chords are instruments. Playing the vocal chords well takes as much skill, talent, and years of practice as any manmade instrument.

We have the same semantic problem with the words, artist, painter, photographer. Does that mean anyone who has drawn a stick figure, finger painted in kindergarten, or pushed a shutter release is an artist, or painter, or photographer?