Is there a "name" for price leadership oligopoly? Oligopoly equilibrium in the static setup consists of cournout equilibrium and bertrand equilibrium (Nash Equilibria)while in sequential games we have stackelberg equilibrium and price leadership equilibrium (subgame perfect nash equilibria).

my question is why is there no name attributed to price leadership equilibrium? Is that just how it is?
Oligopolistic competition through both quantity and price can be regarded as Stackelberg competition, provided that there is a leader firm that moves first (commits) and other players move subsequently with some information on what the leader has committed herself to. It thus describes the dynamic form of the classic Cournot and Bertrand competition.

Some explanatory background info:
Stackelberg first published his sequential move games in 1934 under the title 'Marktform und Gleichgewicht (Market Structure and Equilibrium)" (I unfortunately could not find a pdf of the paper, not even in German). And it seems that this sequential nature was indeed ground-breaking, in this case. At this point Cournot and Bertrand competition were already well known concepts, but game theory was not an established field yet ('Theory of economic games and behavior' for instance was only published in 1944). This first model apparently focussed on competition in quantity. But as this paper shows, price competition was not an unknown concept to him, either (the paper is in German but there is a English summary on page 139).