phrase requests - Word that means "private word used by a group"

Here's an example: a group of my friends has a term we invented. It's a verb that means "to pronounce over-correctly", used of a word borrowed from another language when that word has a common anglicized pronunciation, but the speaker of the word chooses instead to pronounce in the source language. There may be a standard English word that has this same meaning, but our private word is derived from the name of a person we know who does this often, and so it's sort of an inside joke: "Oh, stop Trauthing, you pompous fool."

The word I'm looking for describes such a private word. Sort of like a single word in an idioglossia but with no surrounding language; the private word stands on its own. Taking a cue from neologism I might coin the word idiologism but that word apparently already exists (or so says google) and has a slightly different meaning since it's only used by one person.

Is there a word for a coined, private word, which has a specific meaning only to a certain small group of people?

Answer

I would call the word itself a pet term. This is an interesting topic, and pet terms are probably common within families and other small, tight-knit groups.

An example of this usage, in a headline from a media analysis website:

Limbaugh Explains His Pet Term "New Castrati": Men Who Are "Bullied By Women And The Power Structure And Liberalism"

Note that this is not the same as a term of endearment.

EDIT: This is not an established linguistic term. I consulted a few sociolinguistics textbooks and didn’t find any discussion of this kind of ephemeral in-group language. One place to look would be studies of college slang, like Connie Eble’s Slang and Sociability.

Pet term is also different from pet word, which seems to have a generally accepted meaning: a word that is frequently used or otherwise favored by an individual person or writer. Pet words are already part of the general lexicon, though. Brad Leithauser has a 2013 New Yorker piece about pet words. He gives some examples (sweet for Shakespeare, lad for A.E. Housman) and compares them to stray cats taken in by their users:

Each of these words presents the critic with a little puzzle of devotion: What was it about this particular package of syllables? Why was this stray cat escorted into the author’s studio and offered a saucer of cream and a plump pillow by the fireplace? It’s not as though the studio were soundproof; during working hours, the author no doubt could hear other strays, seemingly no less deserving, meowing clamorously for admission.

Link: Pet Words

So, pet term (rather than word) can convey that it's favored by the in-group, but not established as a true word yet.