It seems like you’re not being clear about how you are thinking about the cases, or are misusing some of the terms. Nash Equilibria exist in zero-sum games, so those aren’t different things. If you’re familiar with how to do game theory, I think you should carefully set up what you claim the situation is in a payoff matrix, and then check whether, given the set of actions you posit people have in each case, the scenario is actually a Nash equilibrium in the cases you’re calling Nash equilibrium.
It seems like you’re not being clear about how you are thinking about the cases, or are misusing some of the terms. Nash Equilibria exist in zero-sum games, so those aren’t different things. If you’re familiar with how to do game theory, I think you should carefully set up what you claim the situation is in a payoff matrix, and then check whether, given the set of actions you posit people have in each case, the scenario is actually a Nash equilibrium in the cases you’re calling Nash equilibrium.