You might follow up with the diagnosing dr. to make sure there are no crossed wires; why was the written report so different from the conversation? It's very legitimate to ask directly.
How are your DS's social skills? When he's with a peer, can he have a meaningful two-way, multi-turn conversation? The social impairment is one prominent thing that would differentiate a gifted kid on spectrum from a gifted kid otherwise; the non-spectrum kid can have a real conversation, even about topics that they don't obsess about, whereas the spectrum kid might only be able to have the conversation about their special interests, or introduce the interest into every conversation. The rigidity is a hallmark, too, but not a completely defining one IMO.
You might want to get a second opinion from someone who specializes in kids on spectrum; they might be able to tease out these issues more thoroughly and figure out what your kid needs. You may also be able to get services from your district (PT/OT/speech) even while homeschooling.
Dee
