"Superman is what I can do; Clark is who I am" often seen as a pivotal point of the reinstatement of modern Superman's humanity as central to his character further distancing Superman from the reality and Clark as the truth. But personally for me beneath this all I see is a fairly poor read of the human race from a dull guy who'd rather be doing the easy thing rather than something that requires hard work and dedication.
One, your name is just a title given to you by your parents and you as a person have no input on what it is. The choosing of the Earth name over the alien or superhero name has become a kind of weird point of fixation for people that really take comfort in the "human" take on Superman but I've always seen the name game as nothing more than semantics. People change their names all the time, choose to go by nicknames, middle names, or last names. Clark had no more control over the name Clark, than he had over Kal-El, and depending on the continuity Superman.
However what you do is a conscious decision and an outward show of the person you are on the inside. The way I see it what you do IS who you are hence the "Actions speak louder than words," quote. If we were to lay out all the content ever produced of Superman I'm pretty certain the greatest chunk of it would have him in the tights doing the Superman thing.
I feel like a lot of modern work has been kind of obsessed with trying to make sure Supes doesn't get too uppity, trying to cement the farmboy thing as central and trying to strip him of anything that might seem Super. The Kents are put back in to "keep him humble" despite the fact that the guy has consistently been one of the most humble, respectful, and giving people his entire existence. If the guy so much as attempts to stand up for himself he's seen as a "jock" or a "bully" or some other kind of nonsense like that. We accepted things like the idea that Superman wasn't the first hero and that the golden age character pretty much started things and Superman was more the first of his generation. I can accept that but it seems like once you give up one thing people are constantly pushing for more. Then this new Superman animated looks to be trying to say Martian Manhunter and Batman also predated him along with who knows else. Like wise you've got modern writers trying to say Supergirl is better at the hero gig than him and that his half human son is stronger than he is.
IDK and it's probably just me but it feels like the general point of humanity in the modern Superman is to handicap him and hold him back.