"Objective" is one of the many perspectives by which we structure the world that we know in order to express it. "Objective" is the perspective that expresses truth about things. You own that perspective, and you can use it anytime you need to express truth about things. You also own "subjective," and can use it to express a truth about how you feel about things. Both are useful, both are tools available to you.
Neither is false.
I still don't think you understand me when I say "the existence is objective"
Objectivity is the truth about things because it expresses that. Subjectivity is, as you even said, how one feels about things. We feel something about something, but just because we feel that way about something doesn't mean it's true, and the fact that other people feel differently towards that something proves it is not true.
The fact that you feel that way about something is objective and true. It isn't subjective that you feel a certain way about something, it is objective because it expresses the truth about things, about how you feel about it.
So the fact that you feel that way about something is true and is objective. The feeling isn't true and subjective.
Because if there was, we'd believe that there was, and wouldn't that make it "subjective"? (the way you use the word)
I'm not using the word any differently than you, I think you're just not understanding what I'm saying.
I'm not saying belief is subjective necessarily in all cases, just that the way you perceive something is subjective.