You are correct that in AmE we use take more than get to mean use as a means of transport.
But, there are circumstances where each is correct and/or incorrect.
You take the subway (underground), a train, a bus, a taxi, a boat, etc.. Generally, these are situations where someone else is doing the driving.
While you are on the vehicle, you are taking it. It is a continuous process.
If you say I took my car it means that you brought your own car somewhere, as opposed to someone else driving you. In other words, you transported your car somewhere in the process of it transporting you. The difference is subtle, and some might argue non-existent.
But, in AmE it sounds a bit funny to say:
Take your car to New Jersey, I'll meet you there.
Whereas, you would definitely say
Take the train to New Jersey, I'll meet you there.
For the first example, most would say "Drive your car to New Jersey . . ."
Now, as to get in AmE:
Get carries the implication of boarding the vehicle in question (and in most uses is interchangeable with catch -- e.g. catch a cab, etc.).
Americans get a taxi, get a bus, get a ride, etc. But, the implication is that you get things like the train, subway, taxi, bus, etc. at a station or other point of embarkation. In other words, once you are on said vehicle, you have already gotten it, and now you are riding in it.
If you say get the car, you mean, let's go walk over to it, and bring it here. The usage is similar to that of take.
Unfortunately, I cannot find any references beyond my word as a native speaker. The dictionary definitions are not enough to explain these nuances.