Your first example ("would rather study") and the other two ("would rather that") are basically the same thing.
When you use the original "would rather + Verb," notice that the Subject of your sentence and the doer of the action have to be the same:
Ex. Beth would rather study
But if you wanna flexibly talk about a different person doing the action, you need to use another structure. And this is where "would rather that" comes in.
Ex. My parents would rather that I...
I'd rather that you...
Also, normally we would use the Past Tense of the Verb when we use the "would rather that S + V" structure.
Ex. My parents would rather that I drove slowly.
Although we use the Past tense, the meaning isn't Past but is still Present.
Ex. I'd rather you didn't tell her.
I'd rather he moved to another school.
I suppose though, about Present Tense Verbs in "would rather that S + V," that the expression is evolving - like what was previously mentioned.