Is there more than a variety of Alzheimer's disease?

No, but... There are multiple causes of dementia. Alzheimer's disease is just one of those causes.
Mixed dementias. Older folks are susceptible to strokes, which can cause cognitive problems (lacunar state), and not uncommonly, they also have alzheimer's (mixed dementia). Also, some alzheimer's patients have vitamin deficiencies, such as thiamine or b-12, so comorbidities must be considered.