You got the first part right about washing with STP. The answer is
no. Two coats of the new paint will cover better than 1 coat of prime
and 1 coat final paint. If you prime, you might be painting it 3
times.

Mick wrote:

I'm going to paint a ceiling (standard 'ceiling white') that hasn't been
painted in roughly 10 years. It's not really all that dirty for being so
old, so I'll probably just wipe it with STP or the like. Should be no
problem with new paint adhering to the old. But here's the question... I
want to be able to cover with the least number of coats possible. Would
priming it first be likely to cut the total number of coats (including the
primer coat) needed? TIA.
---Mick