"Sit down" vs. "sit up"

You would still use sit down no matter how high up they have to climb to get onto the chair. You want them to sit in the chair not stand on it or swing from it or any other things children like to do on a chair other than sit on it.

Get up on that chair and sit down.

or

Sit down on that chair. (no need to let them know they have to climb up to do it)

You also want to be explicit to avoid the inevitable case where the child has climbed up onto the chair and is now spinning around while doing a handstand and when asked why they are not sitting down in the chair they will reply with You told me to get up on the chair, you didn't say I had to sit down on it.

Probably different for adults, where sit up there would be OK to mean sit on the large green chair. See Large Green Chair Minnesota

"Would you please get up on your chair?"

I wouldn't say anything different to a small child (or anyone else) than this. You're trying to force a square peg into a round hole, to use an old expression.

It's like asking whether you would tell the child to eat his breakfast or eat his lunch when you're serving dinner. The only answer here is that in the situation you've given, there is no such choice between “sit up” and “sit down”. People simply do not use either phrasing, at all. This site is about actual English usage, not how it would work in nonexistent, hypothetical situations.

'Sit up here sunshine, (whilst patting the chair with your hand)', or 'sit up on your chair'. But everyone would have their own way of saying this. There is no correct version, though some incorrect ones.

If the child had got as far as standing in the seat of the high chair, I might say 'sit down' but normally I wouldn't say 'sit down' to a toddler unless I wanted them to sit on the floor.