Down 4 b informal : to or toward a place that is thought of as below or away from another place
A1. She drove down to our house.
A2. Come on down and see us sometime.
4 c : to or toward a place that is away from the speaker
A3. He is heading down to the store.
A4. Would you mind moving (further) down so that we can sit here, too?
Up 4 b informal : to or toward a place that is thought of as above or away from another place
B1. She went up to the cabin for the weekend.
B2. How long will it take to drive up (to their house) and back?
B3. Come on up and see us sometime.
5 a : to or toward a place that is close to someone or something
B4. I walked up to her and said “hello.”
B5. Please pull up a chair.
B6. He lay down and pulled the covers up.
I find these definitions very interesting. It seems using down or up makes no difference to the examples in both 4 b when altitude doesn't need to be taken into consideration. Is it true?
However, as for 4 c and 5 a, I might think it's a little bit different.
For A3, I suppose both up and down work just fine and make no difference in meaning.
For A4, I suppose using up wouldn't make sense as moving-up only helps to make things worse.
For B4, I have a gut feeling that using down isn't as good as using up, but I cannot explain why.
For B5, firstly, I don't know whether the speaker is asking the addressee to pull the chair to the speaker or the addressee or either of them depending on the context. Second, I suppose using down would mean the opposite - being away from the speaker; the same is true of B6.
Is my understanding correct? Please help to clear up my confusion.