"My cat jumped on the chair" means that the cat jumped up and down on top of the chair. The same thing goes with "jump on the horse". But, why is it okay to say "jump on the bandwagon" and not " jump onto the bandwagon"? Unless the idiom describes the movement of jumping while the person is on the bandwagon, which doesn't really seem to correspond the actual meaning of the idiom.
One of the reasons that I am confused by this is because this comment: https://forum.wordreference.com/threads/on-and-onto.202558/
Onto generally describes the action of movement onto another thing.
Please put the plates onto the table.
My cat jumped onto my chair.
On can also take the second meaning, however, one must be careful of its use.
Please put the plates on the table. This means the same as Please put the plates onto the table.
My cat jumped on my chair means that the cat is jumping up and down on top of the chair. In this instance, we must use onto if we want to describe the movement from the floor onto the chair."