Before answering your question, the Japanese Wikipedia article on アジア you linked to DOES talk about how the Japanese language defines アジア; it can't be more authentic a definition of アジア in Japanese when a popular encyclopedia written in the Japanese language and intended to be read by Japanese people talks about アジア. I think what you want to know is what アジア and アジア人 typically refer to in a colloquial Japanese vernacular. Or maybe it's more like how different the precise definition you linked to is from the actual meaning and usage of the word in everyday language spoken by the hypothetical average person in non-educational, non-political, and non-scientific context.
The answer is, of course, "depends." But generally speaking, it's closer to what would pop up on your average American's mind when they hear the words Asia and Asians in normal everyday conversation. So, when you talk about アジア, Russia is often excluded, although it hugely depends on context because, for example, Russia and Japan are next to each other so there are many natural contexts where part of Russia just feels like Asia, which it is. Every educated person knows Middle East is technically アジア, but if you talk about アジア人 for example, it typically means East and South East Asians. India and the surrounding region are a bit ambiguous. You shouldn't be surprised if the speaker includes or excludes the region without specifically mentioning it.