I can't help you with much concerning the arcade version. Consider yourself a lucky person for seeing it in action. The sight of the thing compared to my saturn might cause too much emotional stress.
Two lap championship mode is very simple. Like the regular championship your game starts with a qualifying lap and then the race spans across all three of the courses (Beginer, Medium, Expert, I don't know about the extra course. It might a saturn only course.) Now all you have to do is run the 2 laps in the required amount of time on each course. After that the time from all your laps (excluding qualifying lap) are added up giving you a total time. Being an arcade machine the amount of laps in championship mode are the set at the arcade owners descretion. I am assuming that your machine is set somewhere from 3 to 5 laps a course. That would explain you difference in times compared to the ranking area.