Alzheimer's is, by definition, dementia with associated amyloid plaques in the brain. Since you can't detect the plaques without cutting into the brain, the diagnosis is normally given based on symptoms of dementia (significant loss of memory or other cognitive functions) without other clear reasons (no evidence of vascular events, head trauma, brain tumors or other neurological diseases etc).

My understanding is that amyloid plaques can actually be seen with a specialized PET scan now, so it can be more definitively diagnosed in living people.

You mentioned amyloid plaques. What about tau tangles? I thought Alzheimer's required both. If someone (or some dog, for that matter) has amyloid plaques but no tau tangles, is that Alzheimer's? If they have tau tangles but no amyloid plaques, what is it?

And what about the brains that show amyloid plaques, tau tangles, and Lewy bodies? Or plaques plus vascular lesions? At autopsy, most elderly brains show mixed pathology. Does that person have Alzheimer's plus Lewy body dementia plus vascular dementia? Three diseases? Or one brain failing in multiple correlated ways that we've artificially carved into separate categories?

It sounds like we have at least five different pathological markers that correlate with cognitive decline, often co-occurring, with inconsistent symptom mapping. What makes 'Alzheimer's' a disease rather than a region we've named in a high-dimensional space we don't really understand all that well?