1) If I were boarding a train while talking on my cell phone, I would say
"I'm getting on the train now."
2) If I were describing the behavior of a crowd of passengers, I might say
"Everybody is rushing onto the train."

I think the difference is emphasis. Sentence 1 simply implies that I will soon be traveling on a train, whereas sentence 2 focuses more on the physical activity of boarding the train.

That, at least, is how it seems to me.

I came to the tentative conclusion that "on" (like "in") is used when the verb is normally used only in the sense of "movement to" (with little other semantic content - apart from vertical/horizontal orientation), e.g. "place", "put", "get", "stand" (transitive), "lay", whereas "onto"is used with these verbs only when this aspect (especially: direction) is emphasized - but also with verbs mith more semantic content that can be used for activities that do not necessarily imply such movement), e.g. "rush", "climb".

So I would say:
- I put/placed/stood it on the table.
- I got on the train.
but
- They rushed onto the train.
- He walked onto the pitch.

However, I might say
- While the police were getting onto the train, the criminals were getting off.

I just speak BE, but SD3's "physical activity" resonates with me. When I hear "I got on the train" I just imagine someone being on a train - someone who wasn't on it before. When I hear "I got onto the train" I imagine someone climbing up through the door, putting their feet on the steps.