on the way / on my way

Hello. Which is better?

I was caught in a shower on the / my way.

Thank you. kenta

In the short example you provide both are perfectly all right.
However, the two versions do have a slightly different focus: "on my way" suggests you were alone.

There's also "we're on our way" or "are you on your way?" or "is he/she on his/her way?" etc.

Thank you very much!