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Can we use the term "Cournot Game" to describe a competition with output quantity in general? Or, does "Cournot Game" only refer to the game studied by Cournot? For example, if we study a dynamic competition on output quantity, but with very different price functions and some added constraints, can we say it is a Cournot game?

Similarly, can we use the term "Bertrand Game" to describe a competition with price in general?

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You can safely treat (and name) a game in quantities, whether static or dynamic, as a Cournot game. Opting to do so is not wrong, nor inaccurate. A game whose strategy space consists of profiles in quantities was first studied by Cournot in the 19th century, hence he deserves some credit.

Some people use the term "Cournot-Nash", i.e. by treating equivalently the "Cournot" solution of a game to the Nash solution of a game. Evidently, a Nash solution is a broader concept - the equilibrium in Bertrand (price) competition is a Nash equilibrium of the game.

One of the most famous examples of a dynamic Cournot game that is both rich and very easy to read is the uber-classic paper by Levhari and Mirman (1981) (see also this version that uses the Benhabib-Rustichini trick to obtain a closed-form solution).

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