Whether the cap was on or loose is actually irrelevant. The bottom line is they screwed up and from the sound of things the owner and service manager are sharing the same bunk bed so as to keep their story straight.
When coolant heats up it will expand and the cap is designed to maintain pressure on the system to hold that expansion down and raise the boiling point of the coolant. With the cap off the coolant is going to run over in dribs and drabs.
With the engine cold on your car right now you could remove the rad. cap, start it up and allow it to idle, and soon the coolant will start running over into your driveway. This is true of any car.
This service manager has no idea what he’s talking about (a common malady in their world) and if I were you I would consider talking to an attorney and have the att. fire off a strongly worded letter for a start.
Also take the advice of Shadowfax and send a politely firm letter to Honda’s regional office outlining what happened and your intent to sue the dealer and possibly name corporate Honda as part of that suit.
Corporate Honda cannot legally make the dealer stand behind this but they can apply some heavy pressure to make things right.
What a bunch of weasels. Just for hoots, ask the service manager to appear on this forum and try to lay that BS out to some mechanics. The SM is dead wrong and is either lying through his teeth or is an ignorant, incompetent buffoon. The latter I suspect based on the ones I’ve known.