I always say that I'm autistic - and there's logic to it. When somebody has cancer, you can think of them as the person they were (ie healthy) plus another thing (cancer).
Autism is entirely different - it's impossible to separate the person from the autism, because it's been a fundamental part of their personality, perception and behaviour since birth. It's not something that can be medically treated (in any meaningful way) or removed, or separated from their "normal" characteristics in any way. It's always been a part of them, and it always will be.
It's a part of me, therefore I am autistic. I'm built differently, and I'm cool with that (although knowing about it sooner would've been nice). I get that some folk don't like it, though, so I'm not exactly militant about it; each to their own, this is just my choice because it makes the most sense.
As for the OP..."on the spectrum" has always seemed to me to imply a fundamental misunderstanding of what "spectrum" actually means; it often goes hand-in-hand with somebody who says "we're all on the spectrum somewhere", as though it's a linear scale with "not autistic" at one end and "Rain Man" at the other. The reality is that it's more like a 2D colour palette as you'd see in Photoshop or similar, with each colour representing a neurodivergent trait.
Accordingly, while "on" implies a line, "in" implies presence within a bounded shape. If somebody said their friend was "in the spectrum", I'd be a lot happier.
But then...I'd be much happier if people stopped treating it like a dirty word to be avoided, and just bloody-well say "has autism" or "is autistic".