Maintenir is used exclusively to mean that you still claim something you said is true. It is an affirmation you made about an event mostly relative to you, like “I wasn’t there that day”.

Soutenir does not necessarily imply you supported the statement before, and is often used with an idea, an ideology. It’s more philosophical than factual.

Maintenir has a lot of meaning but the general idea is “not moving“, in your case it precisely means not changing your point of view.

Soutenir is more difficult, the concept is “to be with” or “to help“, in you first example the idea is “don’t you agree about what I said ?”.

Your first sentence is a bit peculiar because we don’t use it anymore in current language, just in movies presenting old stories.

For your switched sentences, the second one is false, because your not by the side of someone or something. Unless, like said in comments, the two “Il” are two different persons, otherwise I personally don’t recommend the usage.

The first one is correct, in the idea “you don’t want to change your point of view ?”, but still we don’t use it that much.