The following is copied from this godiscussions.com post by Robert Jasiek for reference.
1) A ko consists of two or more connected intersections.
2) The intersections of a ko can be of one, two or three of the types basic, local or global.
3) A ko consisting of exactly two intersections is the ordinary basic ko with its basic ko intersections and a possible cycle of exactly two successive plays on them.
4) Local or global ko intersections are identified by one or several cycles with possibly more than two plays or passes.
5) Most kos with more than two intersections or long cycle kos can be identified by having local ko intersections - while purely global ko intersections are rare.
6) Any cycle defining a local or global ko intersection must be related to it and the current position in that it occurs.
7) In case of a local ko intersection, a player can construct a cycle whenever being prevented from improving the area score on the cycle's intersections.
8) In case of a global ko intersection, a player can construct a cycle whenever being prevented from winning under the given komi.
9) Example diagrams of kos, local or global ko intersections can be seen in my paper or on http://senseis.xmp.net/?KoInGeneral
10) In a given position, the kos are the same regardless of which rules of play are being used. Different rules of play might lead to different correct strategy though. Some kos are currently worth fighting under some rules of play or not currently worth fighting under other rules of play.
11) The ko definition applies to any possible position under every possible rules of play.
Suggestion:
7) In case of a local ko intersection, the opponent can either prevent the cycle, or prevent local area improvement, but not both.
8) In case of a global ko intersection, the opponent can either prevent the cycle, or prevent the player from winning the game, but not both (note: this requires knowledge of the value of komi).
Whichever. If you put some such texts to the parent page, I suggest that you also use diagrams for every point to explain it also visually.