Some clause structure about “SOURCE said that CLAUSE”?

Solution 1:

Both varieties have the same syntactic structure. The difference is lexical, semantic, and pragmatic.

Mental process verbs like believe and think take complements describing the mentation. They don't say much about the truth of that mentation, as you point out.

However, a special type of mental process predicate doesn't simply describe a proposition; it presupposes it. A presupposition of a sentence is something that must be true in order to understand the sentence.

These predicates are called Factive predicates, and know, realize, and discover are examples. Thus, John knows that France is in Europe presupposes that France is in Europe. And so does John doesn't know that France is in Europe. What John knows is irrelevant to what the facts are.