First I am surprised you knew that the Joker was actually several different people in the comics, however they all bear enough similarity that discussing their individual identities is not particularly useful to differentiate them.
This version of the joker appears in the original comics of course, as well as games like Injustice, the Arkham series, as well as movies like The Dark Knight Rises and cartoons like Justice League Unlimited
In the case of a character with multiple versions it is most appropriate to refer to each by the medium they first appeared in, hence why the original Joker is referred to as the comic book version.
There is also a third version worth noting that tends to be less sinister and more humorous, as well as focusing on simpler issues whenever he’s not simply a tool for Batman to look cooler due to being aimed at children. This lighter version is shown in things like the LEGO Batman movie, or Batman the Brave and the Bold