The definitions of "paycheck-to-paycheck" are different in different surveys.
For example, in the second one you linked the question asked was something like "How difficult would it be for you if your paycheck was delayed by one week?" and they took anybody who said more than "not difficult" as "living paycheck to paycheck". My definition is more strict.
There's another one I remember reading at some point (it was definitely older than these, but I don't remember where it was from) where the question was something along the lines of "do you use a credit card for monthly payments" with the assumption of a "yes" meaning they couldn't pay it otherwise and were paycheck-to-paycheck by necessity. It completely ignored the people who use credit cards for the protections or convenience.
The Forbes article linked above shows that 80% of Americans have $2,000 or less in savings. That's 80% of everyone, not just people living paycheck to paycheck.
That directly contradicts your claim that the majority has 3 months of emergency savings unless you're so out of touch to think that $2,000 is enough to last people 3 months.
These are self reported numbers, and contradict the numbers from the federal reserve, individual banks, and credit card companies. My main problem with the claim is this. If credit were required by the majority of people, how can it be the case that fewer than 50% of people maintain a balance on any card?
For example, in the second one you linked the question asked was something like "How difficult would it be for you if your paycheck was delayed by one week?" and they took anybody who said more than "not difficult" as "living paycheck to paycheck". My definition is more strict.