This question is prone to a wide range of answers because physicians tend to address the question based on a strict, clinical definition of Alzheimer's disease, whereas scientists adhere to a different definition. The rest of the world, including journalists, tends to consider the disease more generally, as explained below.
The scientists tend to think in terms of disease pathology. In their minds, as soon as some biological process initiates a cascade of other processes that will eventually lead to Alzheimer's lesions in the brain, then the disease has begun.
The physicians are bound by the duty of consistency to use the definition published and accepted by their peers. Since the clinical definition of Alzheimer's disease currently requires the presence of "dementia", then physicians peg the start of the disease at the point when the patient has accumulated so much brain damage that they can no longer care for themselves without human assistance.
The journalists and publishers are adept at gathering facts and packaging them into public messages. Being unaware of these diverse definitions, they often repeat "expert testimony" without realizing the nuances therein and without clarifying the context of their statements.
The public tends to think about the disease in terms of symptoms. That is, if one has memory loss or other cognitive difficulties, perhaps there is a disease at hand affecting their brain. If one has no symptoms then they are probably in good health. This perspective would consider the start of Alzheimer's to be at the onset of noticeable memory loss. That happens much later than the scientific perspective (biological change not yet manifest in symptoms) but much earlier than the clinical perspective (severe symptoms meeting the definition of dementia). So what's the answer to the question? It depends on who you ask. The underlying biological changes probably begin decades before early symptoms appear. From the onset of the subtlest symptoms, there is generally a 5-10 year period of mild cognitive impairment during which time an Alzheimer's patient maintains fairly high function and can live independently. This is also considered to be a period when treatment may be optimally beneficial so early detection of symptoms is important. Eventually, the symptoms worsen and the patient becomes demented with an average life expectancy of about 7 years from that point forward.