I think your issue is largely wire size in your electric supply. You got it running by going to a larger wire gauge in the extension cord and reducing its length. Both changes allow more amperage to the A/C. It needs "20-amp service" to run reliably. That doesn't mean just the breaker size that feeds the outlet. It means ALL wiring and ALL connections.
The A/C may have taken time to start for two likely reasons:
1. It has a Compressor Start Assist device (aka Hard Start Kit) that uses a PTCR device to drop the compressor into run mode once it starts. Like releasing the key in your car from Start to Run. If the device is in fact there and in fact uses PTCR, the PTCR has to cool before it can provide start assist again. That's why we don't recommend the least expensive kits.
2. Compressor had already made a number of start attempts and gotten hot enough to activate an internal hot compressor shut-off. Once that happens it can take at least 10-mins to cool and close the power circuit in the compressor. By then a PTCR device would have also cooled, and you'd stand a good chance of a clean start.
The Start Assist issue is why I asked for pix of wiring diagram and the capacitors behind that metal cover.
EDITED TO EMPHASIZE:
That shore tie cable as pictured above IS 120-volts-AC, even though it looks massive enough to make people think it's 240-VAC. Safe enough to assume that a campground pedestal socket for this cable is 120-VAC. BUT BUT BUT people (including licensed experienced electricians) HAVE (actually often) installed outlets on RVers' homes and set them up 240-VAC. It's 120!!! Just massive because it's 30-amp.
Use extreme caution installing an RV outlet OR having one professionally installed. One way to tell is that only a SINGLE breaker is added to the master panel. NOT a paired set like you see on Range, Drier, Water Heater, etc.
