I don't know if the "run crank" relay underhood is the same function as the "starter" relay. I think it's likely the starter but you might find a similar relay under the hood for e.g. the cooling fans and borrow it to swap with the starter relay to see if that works to diagnose the problem.
I went to autozone to do similar checking on a civic recently and the relay cost way too much. It was the starter problem. Same symptoms but hoped a relay would fix the problem.
Edit: Oh and "extra battery" if it's the same won't help. It needs to be higher voltage like jumper cables on a car that's running or a Jumper battery that's higher voltage e.g. near 14v not another 12.6v or less.