Your basic idea is okay. "In time" is not personal, though. It means that something happened at a time that did not interfer with another previously scheduled things.
I am supposed to meet you outside the theater at 7:30. The show starts at 8.
If I show up at 7:50, I am not on time. I did not arrive when I was exected. However, I still arrived in time to see the show.
Let's say you told me to meet you at 7:30, but you read the Web site wrong and the movie we wanted to see starts at 7:20. If I show up at 7:30, I am on time, but we are not in time to see the movie.