From what I read,
"on time" = Opposite of "late".
The right hour, the puntual hour.
"in time" = Opposite of "too late".
At the right time to do something.
In my language (French), it would be translated with
on time" = à l'heure,
and "in time" = à temps, au bon moment.
"in time" = in tempore/tempore
source: wiktionary fr:
Moment favorable, bon moment, occasion, opportunité. à temps, à propos, au bon moment. = in time.
Dicolatin ("au bon moment", = at the right time) = in tempore.
The Gaffiot dictionary gave "Tempore" (without "in") and "Tempori"=at the right/appropriate time (au moment opportun) = in time.
In French, you can also use "à l'heure" to say figuratively "the right moment", but the first meaning is the right hour,
maybe the same with "in tempore".
but if you refer specifically to a time (hour), meaning at the punctual hour.
maybe "ad horam".
ad horam venit = he comes at the agreed time/hour. (on time)
ad horam mortis destinatam = at the hour set for the execution (on time)
"On time" = in tempore/tempore/tempori, and "ad horam"
"In time" = in tempore/tempore/tempore
Other useful expressions:
In tempus: For a limited time, temporarily.
Tempus est: It's the time to...
I've only met the ablative "(suo) tempore" = "at the right time", "at a fitting time", "in time". But okay, it seems that Lewis and Short is giving the thumbs up for this one: "tempus [...] II. Adverb. phrases. A. tempŏrē , and more freq in adverb. form, tempŏrī or tempĕrī , at the right or fitting time, at the appointed time, in time, betimes, timely, seasonably" (http://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0059%3Aentry%3Dtempus)