It's not a big caveat. How else would you define paycheck to paycheck? I'm talking about days, weeks, or low single digit months. A window of time is not indefinite.
The point of measuring people living from paycheck to paycheck is to measure their ability to have financial security. You are financially unsecure if you cannot pay for essentials for days, weeks, or a month off of savings. Many people in the U.S. are completely unable to save up for more than that. That's literally the point of talking about people living paycheck to paycheck. And usually, people only refer to being able to pay for food and housing. It rarely accounts for saving for retirement, health care, home/renter/car insurance, etc., all things that should be considered essential for living but are typically not in these studies.
The point of measuring people living from paycheck to paycheck is to measure their ability to have financial security. You are financially unsecure if you cannot pay for essentials for days, weeks, or a month off of savings. Many people in the U.S. are completely unable to save up for more than that. That's literally the point of talking about people living paycheck to paycheck. And usually, people only refer to being able to pay for food and housing. It rarely accounts for saving for retirement, health care, home/renter/car insurance, etc., all things that should be considered essential for living but are typically not in these studies.