Oh no somebody mentioned "bonded" Long thread to follow! :)
So what does the ground prong hole do in those Honda receptacles in a "two-wire" system?
I did not have the trailer shore power into the Honda during the time the receptacle was showing Open Neutral so I can't say how that would have gone.
The receptacle was working despite whatever "open" it had when it was part of "whole house" with the shore cord into the inverter (inverter not chassis grounded to frame) (converter off, fridge on gas) but when I first hooked up to shore power at home it all worked briefly including the fridge then I saw the A on the fridge and started looking for the problem, back to OP---
OK the house is bonded, trailer not, and I get two lights in the trailer usually same as in the house.
Switching the power off and on using the main 30a breaker in the trailer got it all going again for a minute then it faded back to Open Neutral. It was only when I pulled out the receptacle and "worked" its wires it suddenly began to show two lights again on shore power so I know I fixed it somehow.
I think there was a poor contact in the receptacle that was created when it got "hit" that could not hold power past the turn on but when I worked it, the wire must have gone back in place on its contact in the receptacle more firmly. Just guessing though.
It remains true that the charger plugged into that receptacle's other half was running ok with the tester showing Open Ground in the other half of the same receptacle. I did not try anything else in that half of the receptacle the tester was in. The fridge (several receptacles away in the string it appears) showed A though.
If I had not turned the fridge to 120v and seen that A, I would not have known anything was wrong.