Something being true in a axiomatic system is the same thing as it being provable; that's what "true" means. While a Godel statement for X can be interpreted as "X does not prove this statement", that interpretation inherently relies on the semantic implied by X. The Godel construction is systematic way of generating independent statements without needing to know anything specific about the axiomatic system.
Ah yes, you're right, and my lazy wording in the former comment is inaccurate. A Gödel sentence is just a statement written in the syntax of whatever formal system we're dealing with: generally, it's a statement that there exists no natural number which satisfies a particular property. The formal system cannot prove or disprove that statement.
As you said, we tend to call the statement "true" because we know that the formal system itself was designed with the intention to describe natural numbers and arithmetic, and the statement was designed intentionally to refer indirectly to itself and claim its own unprovability. Since the statement is formally unprovable, we interpret it as being true. I had forgotten that Gödel actually showed that there are other interpretations of the formal system in which the Gödel statement is false.
