Hello Sam. I'd say the subtle difference is one of register ~ board is a bit more formal than get on. A notice on a bus might read Passengers are requested to wait until the bus has stopped before boarding; the passengers themselves are more likely to talk about getting on and getting off
If you get on a car, you are climbing on top of a car's roof.
As has repeatedly discussed in this forum, the use of prepositions in English (as well as many other languages) just has to be learned.
As an added note, your dictionary definition is not correct since "get on" has no particular linkage to a public vehicle.
For example, I get on my tractor to cultivate my garden. Likewise, I got on a cruise ship a few years ago. Additionally, you might say "get on" a truck if you're talking about the open truck bed (often used as transportation in some parts of the world.)