Yes, it was campy, but I was in kindergarten. So to me, then and for many years after, Adam West was Batman.
I hadn't yet discovered comic books, and the Dark Knight was still years away. The show could be called a takeoff on the Silver Age Batman, whose adventures took place within the bounds of the Comics Code Authority and came to be regarded in later years as fluffy and silly. Really though, there wasn't all that much room for parody.
I just happened to be at the right age, completely innocent of The Batman's origin story, to be enthralled by the show. It was only later, I think after the show's original run, that I learned why Bruce Wayne had put on the costume. By then I had bought in to the characters so much that giving me a book containing some of the old comic-book adventures was a pretty good way to ensure that I stayed out from under foot for an afternoon.
Though I was momentarily confused that Commissioner Gordon in the comics looked more like Alan Napier than Neil Hamilton. And I wondered where Chief O'Hara and Aunt Harriet were. That was my first exposure to the differences in how characters were realized in different media, or even in different outings in the same media -- after all, the origin story was definitely pre-Silver Age.
Still, in my head I still always heard Adam West's voice when comic-book Batman spoke. As for later TV incarnations, Olan Soule? Who was he?
It wasn't until after the Tim Burton series of movies that TV successfully replaced Adam West as the quintessential voice of Batman in my memory, when Kevin Conroy took on the role. Having the chance to play off the best Joker voice ever, whoever that guy was who had the same name as Luke Skywalker but couldn't possibly be him, didn't hurt.
Still, Adam West kept going, eventually voicing a Batman-like TV superhero on an episode of "Kim Possible," in between his duties of voicing Quahog's mayor on "Family Guy" (a guy who, according to West, was named Adam West and looked and sounded just like the actor of the same name, but wasn't actually him).
Actors whose later opportunities end up limited because of one definitive role often complain, for a while, about the burden -- but even Leonard Nimoy eventually admitted that, yes, he was Spock. If Adam West ever complained he was low-key about it, and like so many others he found a way to turn the limitation into a spotlight of his own that no one, not even Kevin Conroy, could steal.
The actors playing the arch-villains on "Batman" may have been more famous when the show was on, but Adam West was the star.