I want to make sure everyone understands this difference and the reasons why it is important. Subjectivity isn't a big part of this dialogue, and it isn't here at all. Work it out on paper, and reread the passage you think you are paraphrasing. Quote directly from the text, this may solve some of these problems. But just think about it.
Statements can be true or false.
Therefore Names are objective. The second statement is not implied in the first. Ironically, Hermogenes later agrees that politicians are those who are most adept at naming, which points to a social construction of language. He does this completely intentionally. He and Socrates review some of the points of the discussion.