Michael Swan’s excellent Practical English Usage (Oxford University Press), provides a succinct answer to both of these questions:
"On time = at the planned time; neither late nor early:
 |  | | Peter wants the meeting to start exactly on time. |
In time = with enough time to spare; before the last moment:
| | | He would have died if they hadn’t got him to the hospital in time." p. 450 |
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