I'd say "I spent the whole morning in bed", but "we found the cat in the bed". I think "in bed" relates to persons, and to what they may do in bed, whereas "in the bed" refers to the location.
For B to be correct, you'd need some previous context through which you would have learned that John had thrown himself on the bed. Without any context, as is the case here, the general picture that comes to mind is that John is in the bed.
"Stay in bed" is what I would say to someone, for example in the morning, before s/he's had the time to get up, so to stay under the covers, the quilt, whatever they're under.