If we include the larger research community -- economics, computer science, social sciences, business schools, operations research, etc -- I think there really is a partition between combinatorial game theory and what I would propose to call "equilibrium" game theory. Most in econ and related fields who study/use game theory extensively have probably never even heard of the combinatorial kind! And even those who have usually don't encounter it in their research. (For instance the course you link is by two computer scientists and an economist.)
And I think this divide illustrates the nature of the partition, namely, "equilibrium" game theory is at home in microeconomics: Its purpose is understanding the behavior of groups of strategic / self-interested agents. I think the following motivation helps illustrate. A choice facing a single agent is simply an optimization problem about maximizing utility and hence, once written down, has a well-defined solution. With multiple agents making decisions, however, one needs to propose a "solution concept" describing or predicting how groups of agents might behave. There is no necessarily right or best answer. The insight developed by von Neumann, Nash, etc was to propose as solution concepts equilibria, where the key point of equilibrium is that all agents are simultaneously optimizing. Essentially all game theory of the second kind, in my experience, follows this motivation and solution approach, hence my proposal for "equilibrium" as the descriptive term.
(By the way, for this reason I would argue that "cooperative game theory" is a misnomer. Although it is also taught in the same economics classes, it has little to do with "game theory proper". It fits better within social choice.)
On the other hand, while I have almost no experience with combinatorial GT and probably shouldn't risk putting my foot in my mouth, my impression is that it is not generally motivated by modeling strategic agents. Instead, it tends to use a "game" as an analogy or mental picture for describing a well-defined mathematical problem in which issues surrounding strategic behavior, and especially the problem of solution concepts, do not tend to play a role. The question, although described as involving multiple agents, is more about understanding the (well-defined and uncontroversial) optimization problem.
To highlight this, in my (limited) experience, even in artificial intelligence where problems related to combinatorial game theory come up (planning, alpha-beta pruning, solving perfect-info zero-sum games like checkers/chess/go), the problem is not really described as falling under game theory (which to that crowd tends to mean equilibrium game theory) but rather simply algorithm design or optimization.
So in summary, I'm hoping to put forward two points. The first is that "equilibrium game theory" may be a good disambiguation name for the second kind of game theory you mention. I think the notion of equilibrium actually quite closely capture both necessity and sufficiency for falling into that category. The second point is that, if you look at the broader research community than in mathematics (not to mention popular culture), a large majority equate "game theory" exclusively to equilibrium game theory (and generally out of ignorance rather than conscious choice). I'm not sure what the point of that point is, but maybe it's useful.
Best Answer
I think von Neumann dealt with the case $n=2$, and it was by no means obvious how to extend the concept of equilibrium for the general case and prove that it always exists. More precisely, $n$ players before Nash were reduced to the $n=2$ case by partioning the players into two groups in all possible ways. Once you regard several players as a single player, they are meant to cooperate as they must act like a single player. Nash is very clear about this in his 1951 Annals paper: