I record my students lessons with studio quality recording equipment and then give them the file on CD or memory stick afterwards so that they can practise at home - I don't use the recordings in lessons though.
I do use a lot of backing tracks in lessons, simply because a lot of professionals are used to working that way and they bring their own backing tracks - frequently recorded by their own bands. It makes sense to use them rather than a piano accompaniment.
I don't always give a score for songs we're working on either - a lot of the traditional celtic repertoire doesn't have scores for example, so I use these instances as ear training exercises. If we're working on something that does have a score, then I'll encourage the singer to use it during the learning phase, but to memorise as soon as possible afterwards so that they can move away from it.
I don't think I'd tell a student to warm up with their recording whilst I went to get a cuppa, though!

I use the warm up phase of the lesson as a listening exercise for me - is everything where it should be? Are things moving when they're supposed to? Is the tone connected and free etc. - if not, then that's the starting point of the lesson.