Terminology for "part" of extensive form game
A "part" of an extensive form game that is not a proper subgame because it does not start at a single node but an entire information set would be called "continuation game". This terminology is fairly standard (Perfect Bayesian Equilibrium).

However, I think what you are after is a stochastic game which consists of several states. Each state corresponds to a different game. In your example there would be two states: One state for each player $i\in\{1,2\}$ being the Stackelberg leader. Then you also need a transition process that maps the period-1 state and the period-1 action profile into a probability distribution over period-2 states. You then find equilibria for all state games in period 2 and proceed to solve for equilibria in period 1 with the corresponding (probability-weighted sum of) continuation payoffs.