Think about it... If Gen alpha agrees not to have kids, there will still be a generation after them, raised by Gen Z. A lot of families have a generation gap between parents and their children.
generation (noun): "the living things which share a common ancestry and are alive at (or about) the same time."
The word does not refer solely to humans. It existed before the habit of neatly labeling people born within an arbitrary range of years. For example, medical researchers will record changes in microbes from generation to generation (without making up names for each generation).
So, if our generation (the people alive today, or any time in the next ~9 months) agreed to not have children then humanity's climate crisis would be solved. In fact, every human problem would be solved. Think about it.
I'm not defining terms; I'm repeating how the terms are defined. You can look it up in any dictionary. After that, if you still feel like arguing about it I'm not your opponent. You can contact Mrs. Mirriam-Webster, or Mr. Oxford, etc. Please post the exchange! Maybe the argument, "That's not how it works," will convince them.
Since you mentioned Webster... See definition "1.b" for the most common meaning of the term when people are talking about human generations (Millennials, boomers, anything that starts with "Gen"). Context matters.