Autism Parents and People With Autism

I find it quite interesting that the same people who are so insistent on person first language being used to describe their children (and all autistics, including ones who are adults who prefer "autistic person" or even use "autistic" as a noun sometimes, like me), are the same ones who call themselves autism moms or autism parents or autism siblings. It's... confusing, to say the least, and certainly hypocritical. So, why do they do it?
(This came up because I saw that it was bothering someone else, too. That's actually how a lot of my ideas come up- someone says something or asks something and I go "I need to talk about this. BRING ME MY BLOG.")
Anyways, I thought and I thought. People put identities that they consider an integral part of themselves first, but not identities that they don't consider an integral part or which they don't want other people to see as an integral part, right? That's why a person is a woman or an asian or agender or any of a number of things, but has a broken leg or has cancer. Being female, being asian, being agender, being X is important. A broken leg is something that is transient, yes, but it's also not all that important to who the person is. An amputation is permanent, but as far as I know, people with amputations actually do prefer people with amputations to amputee. Sure, it's important information in some situations, but it does not define who they see themselves as or they do not want it to define how other people see them. Cancer is interesting too. People have cancer, which is unfortunate and generally not used as an identity. People are cancer survivors. So they think that's important.
Which leads to an idea. Autism parents consider the fact that they are taking care of an autistic child to be an important part of who they are and what they do. That much is pretty clear, so wanting to call themselves that makes sense. The ones who call themselves autism parents tend to be the same ones who subscribe to the tragedy model of autism, who want it cured, who sometimes think that autism stole their kid. So the idea that autism is part of who their kid is probably hurts too much to look at, and so the kid is a person with autism. Seeing anyone call this thing they see as having stolen their kid from them a part of who they are hurts. So they are person-first crusaders. Does this make sense? Well, it's not completely consistent, but we're humans. Logical consistency is not one of our strong points. However, it would explain the discrepancy. I will, of course, be asking an autism parent about this, but I don't expect much of an answer. So, what do the rest of you all think? Is it possibly this or am I way off base?