Would rather (but not would prefer to) is also followed by a past tense when we want to involve other people in the action, even though it has a present or future meaning. Study the following:
a.Shall we go out for dinner tonight? ~ No, I'd rather we ate at home, if you don't mind.
b.My rephrased: Shall we go out for dinner tonight? ~ No, I'd rather we eat at home, if you don't mind.
I have extracted the italic parts from a grammar site, but would you tell me if the bold part is correct, too? If so, what is the difference between a. and b. semantically?