I understand the distinction that the website is trying to make, but I don't think it really exists. In the particular example it gives (my wife is asking about the possibility of me cooking a meal for friends, knowing that I may be short of time) Then "Can you cook for our friends on Saturday." seems correct and idiomatic, and has the same meaning as "Are you able to cook ...."
In your example, again I don't see much difference in meaning between "Can you help me" and "Are you able to help me". Using "can" also raises the matter of possibility and not just ability.
Can you help me with my homework?
Not right now, I'm cooking dinner.
It would be odd to use "Are you able to..." in that context, as it is more formal and this is a home-life context, and the child wants to know not only about the parent's ability, but also the possibility. It would be wrong for the parent to answer "Yes I can." and then not try to help the child. The question is not really about if the parent "can" help but if the "will" help.
The main reasons to use "are you able" are 1) to use tense
I will be able to help you with your homework at the weekend, and I was able to help you last weekend.
And 2) to avoid the meaning of "is it possible" or "is it permitted" that "can" implies.
Am I able to still play tennis?
(a patient asking a doctor about their ability to play with an injury, and not asking for permission)
Finally, 3) The "be able" form can add a little formality
I am able to program in Java and C.
(formal writing in a CV or resume)