Sorry, yes, 9 times out of 10 it will start after a few cranks. I posted the video of the worst starting attempt I'd recorded because I thought it would ultimately be more useful in diagnosing the issue, but that may have only confused things.
I want to rule out simple electrical problems first. I went to check the starter relay, but the owner's manual doesn't identify the relays, only the fuses. Diagrams I found online (attached) indicate there are 3 different starter relays. I don't understand why there would be three. But the indicated starter relays are R2 and R5 in the engine compartment and R3 in the passenger compartment.
I pulled R2 alone and the car wouldn't crank at all, which is what I was expecting.
I pulled R5 alone and the car still started in the same difficult start.
I checked R3 in the passenger compartment, and was surprised to see that no relay was even installed there. I moved the R6 power window relay there to test but still had the same difficult start.
Are there really three starter relays? Are they redundant? Or do all three need to be in working order to deliver full power to the starter?