The most common question new users of the trope rename forum had, particularly those who believed that renaming tropes was a misguided practice, was why terms like Brother Chuck or The Other Darrin were considered immune from rename action when other similar character named tropes such as Jonas Quinn or Spikification were dismissed. The main reason for this was simply pragmatism- Brother Chuck and The Other Darrin were terms that had been coined over the course of decades of repeated use in small circles of TV aficionados. TV Tropes aims for a larger audience to be enraptured in a shorter period of time than that- so more intuitive trope names are necessary. This explanation, while accurate, is more philosophical than what most people are looking for when they just want to know the reasoning behind rename policy. So users got into the habit of explaining this favoritism through the Preexisting Term.
The psychology of Preexisting Terms is predicated on the same self-doubting philosophy I've described of heavy TV Tropes users thus far. We were just a bunch of random people on some Internet web site. What authority did we have to arbitrarily name tropes in fiction? TV Tropes has always had mild issues about its legitimacy, and Preexisting Terms were a definitive statement that we were just continuing work that had been started some time ago.
While this term solved the issue of "how do we explain away this one complaint people keep bringing up?" (at least for those who went to the trouble of asking us), it managed to bring up another issue of what exactly a preexisting term was. TV Tropes does not use the concept of notability. So while the phrase Preexisting Term was intended to safeguard terms that had been used in television writing for decades, it was often trotted out for terms that were popular in specific communities but virtually unheard of outside of them. Face Heel Turn and Heel Face Turn were two such tropes courtesy of the wiki's Professional Wrestling faction. The same is also true of many Anime-specific terms like Zettai Ryouiki which generally only appear in Anime to begin with.
Bear in mind that usage of these terms is fairly easily verified outside of the TV Tropes context. Others are in a much more murky area. One trope I specifically challenged on this was The Firefly Effect- a trope which describes fan tendency to avoid certain shows (such as Firefly) under the assumption that said show will be Screwed By The Network. It was impossible to prove then, and impossible to prove now, that this term has ever had any meaningful usage outside of TV Tropes. However, it was believed to originate from the buffyistas web board that supposedly led to the creation of TV Tropes, so it was ultimately left alone- I believe an admin told me this fact, mainly because had anyone else told me I likely would not have believed it. It's hard to prove that even the buffyistas web board ever existed any more.
Oddly there was never any real discussion of what exactly a Preexisting Term was- everyone just got into the habit of using it and nobody even bothered defining it. It's a fairly intuitive term, so this is somewhat understandable. After reading it most people can guess what it means, and it never really occurs to anyone to delve any deeper than that.
Of course, to a large extent, I doubt anyone really wants to. If we had ever tried to actually define Preexisting Term, it likely would have been discovered that many titles apparently protected by it would not have had significant backing, for the simple reason that any actual page by the name of Preexisting Term would have to explain how or whether a title qualifies. Source pages that can't even be found via search engine would beg the question of whether the term ever existed at all before TV Tropes, and whether it was only identified as a Preexisting Term to make the title more difficult to challenge.
Such accusations against a fellow troper's honor would be considered serious- another reason why few sought to have the definition expanded on. For what it's worth, I don't believe anyone has ever been entirely disingenuous when claiming that a term is preexisting. When it gets right down to it, the whole Internet is just a series of factions. A person used to calling a certain concept by a certain name with her friends can be genuinely dumbfounded to discover that not everyone does this, even if the title just seems so obvious. In any event, there's little to be done about the matter anymore- explicit Trope Namers are seldom proposed nowadays, so the Preexisting Term is now mainly a nag about old titles which, no matter how long they've been around, still strike people as bizarre.