That site, and implicitly, your post, are negligent of the fact that things with precise formal meanings often creep into the realm of being figures of speech. When someone says, "It doesn't follow that..." they don't mean, "there exists no proof which verifies A -> B" they mean, "that doesn't make sense".
There is certainly a place for precise terminology, but it's not necessary in casual conversation.
Your example and "begs the question" are different cases. In your example, when people say "it doesn't follow that" they mean one can't draw the following conclusion from what has been said. That's the same meaning as in the technical sense. The only difference is that in a technical context, there's a whole lot more evidence backing up the statement. It's a difference of degrees, not fundamental meaning.
On the other hand, "begs the question" is a logical fallacy in the technical sense; it means someone has assumed what they were trying to prove. It has no relationship with "raises the question." They're not trying to say the same thing.
There is certainly a place for precise terminology, but it's not necessary in casual conversation.