After @tlhIngan re-posted (and sadly shortened) the tweet, @darinarrick replied.
I was going to jump on in, but other things got in my way. I also had more to say than could fit in a 140 character message. So here we are.
Yes, in "The Trouble With Tribbles" (TOS) Korax boasts that "half the quadrant is learning Klingonese". Impressive.
That word in not heard again until Quark uses it in "Looking for Par'Mach in All the Wrong Places" (DS9).
The in-universe novel "The Final Reflection" by John M. Ford refers to the language as "Klingonaase", giving "Klingonese" as a Human mispronunciation.
All other on-screen references to the language simply call it "Klingon". And why not? Star Trek is very good at reuseing the same name as often as possible. Spock is a Vulcan. He was born on Vulcan. He speaks Vulcan.
And then came Marc Okrand. (Well, "then" is a relative term timeline wise...). The language that he created is called tlhIngan Hol. It simply means "Klingon language" in Klingon. And in his book "The Klingon Dictionary" the language is referred to as "Klingon".
So even though the actors seem to be using "Klingonese", the fans and the creator of the language himself, refer to it as "Klingon".
And given the similarities between "Klingonese" and "Klingonaase", especially in the way they're pronounced, I'm more likely to merge them into the same thing. I only use the term to refer to the language as used in "The Final Reflection".
I'll use tlhIngan Hol to distinguish between "Klingonese" and "Paramount Hol" (the language as mis-used by the TV writers).
Otherwise, I'll simply say "Klingon".
If I were to list the language on an application, I would write "tlhIngan Hol".
is how it looks in the native Klingon writing system of pIqaD.
Doing a Google search on "xifan hol" will pull up results on tlhIngan Hol.
The HTML language code for Klingon is tlh.