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Definitely what you say is the accepted meaning. But wouldn’t you say A1 is more correct than A2? Possibly relevant, there’s also

A3: “Sure I do, last time you sat next to me you wouldn’t shut up.”

That's an excellent example to show why statistical NLP outperforms discrete parsing using logical rules. That A1 and A2 means the same is something that is clear to us from the context. This is something a continuous vector space model can capture and a discrete rules based model cannot.

I'm not sure I understand, "mind" means "object to", so they are asking "Do you [object to] me sitting here?";

"Not at all" == "I do Not [object to you sitting here] at all"