Ko is an ambiguous term. It refers to one particular shape, a class of shapes, a particular nature of a fight, the fact that a ko shape is under fight, etc. The most frequent usage is for the 2 adjacent intersections and its basic ko shape. To reduce ambiguity, the nature of a ko should not also be called "ko". Therefore, a new term shall be introduced:
An x-ko is a ko with x approach-moves that one side has to make while for the other side the ko is direct.
So a 0-ko is what was called a direct ko. A 1-ko is an indirect ko with 1 approach move (also called 1-move approach ko or 1-move approach move ko). A 2-ko is an indirect ko with 2 approach moves (also called 2-move approach ko or 2-move approach move ko). Etc.
Teigo Nakamura uses a similar terminology, where K^x denotes an x-ko, except that he still calls K^0 a direct ko. Below the x, he might also attach L (for Black) or R (for White) for the (mathematical) player for whom it is an x-ko.
Hyperpape: "The nature of a ko should not also be called 'ko'." I'm afraid I can't follow what this means.
RobertJasiek: "ko" is the part of the board with potentially repetitive behaviour. The "nature of a ko" is a ko's characteristic of how tactics and strategy develop (Who captures first, direct or with approach moves, important for both or flower-viewing, etc.). If we called the part of the board "ko" and its nature "ko", then we would not know whethere "ko" refers to the part of the board or to its nature. By writing x-ko for a specific nature, we know that we are talking about the nature and not just about the part of the board.
Patrick Traill: What do you like to call a multi-stage ko then?
RobertJasiek: An n-stage ko. The minimum is 2-stage, with which I agree to the traditional naming of 2- or 3-stage kos (I have not seen names for more stages in traditional texts).