If you believe the job fulfills specifications and satisfies
your expectations, by all means pay the man. In my opinion a lack of that second coat will come back to bite you when the paintcures--It takes about thirty days. If he wants full payment ($5000 means nothing to me without footage and elevations), IN MY OPINION he needs to honor the terms stated.
I just finished a job on a lake home, a 1500 (- or +) sq. ft cedar lap-sided "rambler" (oh did it ramble) and detached double garage, ONE coat of satin exterior over flat latex stain (same color, or it would have been a 20% +), including garage door, deck rails, steps and risers, for just under $6 grand. We were persnickity in detail, cuts straight like a knife (regardless of the hack-job of the previous "painter") and included priming and painting a new white vinyl window to the anodized bronze of the rest of the house.
It stopped at one coat because the sheen was showing the ineptitude of the previous painter: Sander marks all over the siding were un-fixable without a $10,000 resand job, and a second coat would have magnified these sins.
I hope I'm helping you, without condemning a paint job I've not seen, but I have a pet peeve for paint contractors who tell you one thing and do something else, telling you they're doing what they told you they would.