Yeah, you're more of a post-Crisis guy than I am. But it's cool my friend, I don't hold it against you.
(JK. I think our opinions actually align quite a bit even if we come at them from slightly different angles)
I think what you say works for that specific Byrne version of Clark, who didn't have powers as a child (except when he did via retcon or writing slip) and thought he was a regular farmboy until high school. Makes perfect sense for that guy.
But for most versions, who had abilities from the start? I think the psychological duality begins to develop as soon as little Clark realizes he can do things other can't, and that it scares them when he does stuff like describe colors they can't see or warns them about the tumor in their lungs they didn't know they had. Any version of Clark who was always different and odd and could see and do things nobody else could has to keep a huge chunk of himself compartmentalized. That's his "normal" operating procedure; to live two lives, and it's the beginning of the "Clark/Superman" duality.
And in that way it's not so different from how you acted differently at school than when you were home, and that continued as you grew up and entered the workforce. Same concept, just cranked up to 11.
So I figure there's always been a difference between "Clark" and "Superman," even before anyone called him Superman, before Clark ever needed two names. The side of himself that we generally recognize as "Clark" is the half of himself that he shows the world, hedged in by the need to keep his other half secret. There's truth to that guy; he *is* shy and quiet and socially awkward. The side of himself we generally recognize as "Superman" spent the first half of his life hiding, hedged in by the need to let his other half live a normal, real life. There's truth to that guy too; he *is* confident and goal-oriented and wise. When Clark put on a cape and went public, that hidden side of his life finally found an outlet, and certainly the dynamic between "Clark" and "Superman" shifted, but I think the tension, the duality, was always there.
For any version that *didn't* grow up like a normal kid, anyway.
And of course Clark thinks of himself as "Clark" since that's the name he's known the longest. But whatever he calls himself in his own head, he's still psychologically split in a way that's oddly relatable, but utterly whimsical....and if you look too deep, maybe just a little crazy.