In the links I posted several comments above, you'll find that it is considered an expense for the purposes of personal savings.
I don't disagree with your definition, btw, it's just not the common one.
1. It does not make clear what the payment is. If it is only to pay the interest portion of the loan, there is little question that it is an expense. That would be outside of the topic of principal repayment, though.
2. If we assume there is a principal portion included in the payment, it is said only with respect to student loans. The product of a student loan retains no value – literally worthless the moment you drive it off the lot. As such, it is not unreasonable to consider the principal an expense. There truly is nothing left. A house, not so much. Unless the house has lost value (an atypical situation), you didn't give up anything.
Seemingly not, else we'd have seen the math already. Also,
1. Not having savings (in the personal finance sense) does not mean one is living paycheque to paycheque.
2. The article you link to defines living paycheck to paycheck as a scenario where the family income does not cover expenses. Principal repayment is not an expense. Outside of the 1-2% with interest-only mortgages, anyone who is paying a mortgage unquestionably is not living paycheck to paycheck under the definition you have given. They have surpluses available to spend on non-expenses.
Which brings us back to wondering why – while maybe technically true – you want to include children in the figures? Again, is there some reason we should be aware of newborns not making enough money to save?
> Try to remain on topic and avoid ad hominem, please.
1. It is on-topic. It explains why multiple definitions are present.
2. Ad homiem implies being directed at a person. The statement is not directed at a person.