I am often asked the question what type of chess player am I, but I am unsure how to answer this question because I do not exactly know what would define each type of player. Interestingly, I am only asked in particular if I am an attacker or a positional player, but surely, there must be more types of players than these. What are the types or different kinds of players? Also, if you could include a definition of each type and include any famous players as examples for types, that would be great as well.
There are two broad-brush categories of players: positional and tactical. Positional players' styles tend to lead them to play moves that choke out their opponents. They are patient and bide their time. There's no better proponent of this type of play than Tigran Petrosian. He's recognized as one of the hardest players to defeat. A more recent example would be Anatoly Karpov, another player that was just so hard to defeat. Positional players keep taking their opponents' good moves away until nothing is left but bad moves. Aron Nimzowitch was the father of this type of play, though he did not invent it - he described and defined the style in publications such as My System.
There are more tactical players. These include pretty much all the darlings of chess - the ones scoring points with fireworks. They tend towards open positions with lots of complexity, relying upon their tactical skills to force an error onto their opponents. Examples include Rudolf Spielmann, a player from another age who made Tal look tame. Mikhail Tal is the more modern archetype - a player able to conjure devastating attacks from seemingly nothing at all. Then there is Kasparov, Fischer, etc the list is rather endless.
In short, tactical players do it to you. Positional players make you do it to yourself.
At any semi-professional level of play, both styles of player can uncork combinational fu. So this isn't to say that a positional player avoids combinations and sacrifices. You won't normally find one player who plays both styles; it is a philosophy sort of thing. They believe one way is best and sharpen their skills to this belief.
I don't think you will get a definitive answer that fits well into the structure of SA sites on this one, but you will find plenty of discussions threads online that cover this topic.
The problem is labels on chess style are open to interpretation, not mutually exclusive, and judged differently at various levels.
You have heard "attacker" and "positional player" the most because they are some of the more general terms and most players fit into one and not the other. Examples of other styles are
I think you could come up with a single list of terms that rarely overlap, then analyze various players to assign tags to them. As mentioned before though, this gets harder as you go up layers. I may think I have a strong endgame, but it is only relative to my other abilities. Sometimes being a positional player just means you are bad at attacking, sometimes it means you are good at attacking but truly excel at setting up position play.
According to the video, a fighter is essentially a practical player, who values chess as played over the board, and isn't afraid to make positional/strategic concessions on the board in order to win. Such a player may purposefully choose a non-optimal way to win a game for the sake of beauty, or the resulting complications it would create for the opponent (i.e. exploiting the fact that your opponent is almost out of time). Nakamura comes to mind as a good example of a modern fighter. Carlsen as well, as he would often choose openings with "dubious" reputation, just to get opponents out of familiar territory. Of course, my favourite fighter would be Tal.
A scientist can be described as a methodical chess player, who would almost always try to find the (objectively) best moves possible on the board. Such a player may spend more time studying chess just to find slight improvements in games/openings/endgames that have already been accepted as "good", and will often employ "textbook" openings/moves in their games, as opposed to "slightly inferior or dubious" (as judged by modern chess engines) moves just to create complications for their opponents. You will find that most chess studies, notably endgame studies, come from the scientists. Petrosian, Kortchnoi and Karpov come to mind when thinking of such players.
Of course, you have a bit of both in most players, but if you were to classify players based on their methodology, then this would be one way to do so.