I'm an architect and my firm routinely specifies interior finishes for projects so I thought I'd contribute a professional's perspective on the issue of how many coats of paint are deemed "acceptable". The fact of the matter is the average consumer usually isn't a paint expert and can't be expected to know about all the factors that impact coverage. That knowledge is considered "means and methods" and in a court of law, the responsibility lies with the painter or general contractor, not the consumer. What the consumer should be concerned about is the final result-does it look good and is it what you expected? The simplest way to communicate this to your painter is to stipule in your written agreement that the number of coats will be "as required to cover". That way all the guess work about what kind of primer, how many coats, how color affects the scope of work, etc., is removed from the consumer's responsibility and resides where it belongs-with the professional. In the contract that's why retention is always a good idea-typically 10% is withheld from payment until the job is completed to the satisfaction of the customer. Of course in return you as the customer have to be reasonable about what constitutes a completed job. Just my $.02.
Consider purchasing supplies personally to save money. Ask the painter for a bid that separates labor and materials. Then explain that you'll purchase the materials and ask for a list of exactly what will be needed to complete the job. Caulking, for example, is an extra supply commonly used to fill any cracks or damaged areas in your walls -- and one that might be overlooked in an incomplete list.
I agree with JHs. If there is no new color on it at all, just talk to the painter about it, if it was an honest mistake (which can happen very easily while painting) your painter should have no problem fixing it. However, in my years of experience, it is not unusual for 2 coats of door paint (good quality) not to cover very well at all. I once painted a red door 7 times, plus a tinted prime coat before I found the door to be a solid color.