I always understand prepositions by thinking of their physical interpretations. "In" describes containment:
"The water is IN the bottle"
"On" describes precise location on top of something:
"The cat is ON the table."
"In time" and "on time" are no different. "On time" describes precise location of an event in relation to time: neither late nor early.
"In time" works just like other "in" metaphors like "in love" and "in trouble". In each case, the "thing", be it love or trouble or time, acts like a container, a sort of kingdom in which you may or may not comfortably live. When you are "in love", you are in the Love Kingdom. What a nice place to be! When you are in "trouble", you are in the Trouble Kingdom, not a good place. When you are "in time", you are in harmony with the Time Kingdom. It doesn't indicate any sort of precision, just that that's a fine place for you to be. So if someone says they arrived "in time for the show", it doesn't mean that the show will start very soon. It only means that time is no problem, as far as seeing the show is concerned.