It seems to me that the expressions in my way and in the way are mostly used in a fairly literal sense, i.e., to refer to some kind of obstruction along a path or direction. For example,
- The protester was in my way, so I shoved her aside and kept going.
- The bulldozer cleared all the debris that was in the way to open up the road.
A secondary meaning of in my way could be 'in my fashion' or 'in the manner that I usually do things', e.g.,
- I scolded him in my (usual) way which made him start bawling.
On my way and on the way also usually refer to a point along a path or direction, but the expressions don't imply the idea of an obstruction as in my/the way do:
- On my way I came across a protester, but I declined to discuss anything with her and kept going.
- Working to beautify the roadway, the orange-clad crew gathered up all the trash that was on the way.
On my/the way can also be used with to to form a sort of complex preposition:
- I usually grab a cup of overpriced java on my way to work.
- On the way to the store, could you drop in on Harry?
Finally, both on my way and--though less commonly--on the way can be used as complete utterances with a meaning that could also be expressed by something like 'I'm (or We're, or It's, etc.) coming now', e.g.,
A: Get your butt over here!
B: On my way.
A: Where is that report I asked for??
B: On the way.