Wikipedia:

Singular they is the use in English of the pronoun they, or its inflected or derivative forms, such as them, their, themself, or themselves, as a gender-neutral pronoun to refer to a single person or an antecedent that is grammatically singular. It typically occurs with an antecedent of indeterminate gender, as in sentences such as:

There really IS a difference, which is the point. Gender neutral means it can be applied to anything relevant, regardless of gender. It's neutral of gender, an identifier that doesn't care about gender. Agender would imply that it's only for things without gender. Wikipedia here also does mention the scenario you speak of, but typically and always are not the same thing.

In addition, as for your "by definition" argument about accurate and correct. Via Google Dictionary:

Accurate: (of information, measurements, statistics, etc.) correct in all details; exact.

Correct: free from error; in accordance with fact or truth.

Accurate is more exact than simply correct. Accurate is correct in all details, so more accurate would be more details that are correct. Correct is binary, something either is or isn't free from error, there is no such thing as "more correct."