Both of you are correct.
Technically, the president can classify or de-classify anything they'd like to at any time. He is the sole and only person in charge of the branch of government responsible for this.
Practically, over many years presidents have delegated this authority all over the place, and in such byzantine ways that most of the folks in compartmentalized projects wouldn't know that he had the authority -- and even if they did, they could just point to various laws and signing letters that make it less than clear how a president would go about declassifying anything.
So yeah, in theory POTUS could take an afternoon off, walk down the street into a government building and declassify whatever he wants. That's why we fight so much over who gets the job: the illusion that these powers are present. In reality, however, if he could get the SS to allow him to take a walk (unlikely), he'd be stopped at the door of wherever he went (both by his own and that department's security services), and then the red tape nightmare would begin. By the time it was all over he'd be better off staying home and watching daytime TV.
For what it's worth, and I hate using movies as historical examples, in the movie Nixon there was a great scene where Nixon went in the middle of the night to talk to protesting students. He wanted the war to end too, didn't they know that?
It was contentious, but then one kid got it and said it aloud, something like "You don't have the power to stop it either, do you?"
Nixon looked completely defeated at that moment. Seconds later he was dragged off by his security staff.
Without getting into the veracity of that exchange, the gist of it is something we see confirmed over and over again in the historical record: large groups of administrators simply cannot be governed and instructed by one person, no matter what the documents technically look like. It's system itself that does the work and enforces the norms, whether it's General Electric or the NIS