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"To be on time" means to do something at the arranged or scheduled time. So if you are on time for a class, a doctor's appointment, or a meeting, for example, it means that you arrive at or even before the time that has been set.
Interestingly, about the method of transportation – usually the train or plane is "on time" or not; the passenger is not said to be "on time." Your sentence would be more natural like this:
"¢ The train was 20 minutes late. I was late, too, so I got to the station in time to get on.
Another example would be:
"¢ I arrived at the airport just in time to make my flight, not a moment to spare. The flight left right on time. _______
"To be in time" means to arrive before it is too late to do something, not necessarily scheduled:
"¢ If you leave your office a little early today, you'll get here in time to take pictures before it gets dark.
"¢ Oh! I really drove fast! I'm sorry I didn't get here in time to see the baby!
"¢ Make plane reservations right away. Grandpa is really sick. I hope we get there in time. He wants to see us again.