As others have said, I think relief is a very common feeling. I know that's what I felt when my ds was diagnosed. I am so glad your ds is in the right school now and is being properly supported so that you and they can see the real him.
As for my ds' diagnosis, we were told he fulfilled the criteria for a diagnosis of Autism Spectrum Disorder. They then went on to explain to us that in order for us to better understand where he was on that spectrum we would probably have heard of Aspergers and previously that would most likely have been his diagnosis, but that most professionals are choosing not to use the term anymore. This was a number of years ago now. As it happens, having done my own research, I don't agree with them aligning his diagnosis to Aspergers, as it simplifies his difficulties and struggles, but then again, that's why the majority of professionals no longer use it.
I can totally understand the op not being able to remember the precise words used. The pressure and stress levels you are under in those meetings are enormous and even when you're expecting it, the enormity of what you've just been told can take you aback to the extent that you don't actually take in the rest of what they say properly. I was warned this would happen and asked if I could record the meeting, which they agreed to.
Not everyone is knowledgeable about Autism, let alone all the technical terms and even what 'Autism Spectrum Disorder' actually means. We all have to go away, take some time to let it sink in and many of us start learning from there. When I think back to the early days of assessments, etc with my ds, I was desperate to learn, but didn't know where to look for the right information and meetings with professionals don't help because they all talk jargon at you and often don't explain things properly. It takes time, you don't suddenly become an expert on Autism because a member of your family is diagnosed.
There is no consensus. There are people in this world that happily refer to themselves as Aspie and don't identify themselves as having Autism, there are others that hate the use of Aspergers and prefer Autism or ASD, others don't mind being referred to as having ASD, but hate to be referred to as Autistic. Some people object to being called disabled, some prefer to say they 'have a disability' others prefer other terms. Often a term one person is happy with, will be the very same term that another person will take huge offence at. It's an absolute minefield and as usual you can't please all of the people all of the time.
For the record. I consider myself disabled. I don't have an acquired disability, I have a condition that is systemic and adversely affects just about every area of my life. That's what makes sense to me, relating to my situation, I do not expect every person I meet or speak to to know that and do not take offence if someone uses a different term in reference to me.