On the general question, I suspect it depends on the magus' personal pride and the nicknamer's social authority, and the House.
In H. Tytalus being given an insulting name by the parens is standard practice. In Ex Miscellanea, a hedge wizard nicknamed by some bully magus may not even realize it's an insult, and might not care what some magical snob thinks. A Flambeau might detonate over an accidental misnaming.
And some Bjornaer magi, need a new, humiliating name for an initiation.
A refreshing change from "my heartbeast in latin"...
I tried to go for the heartbeast in my (Magi's) native language, or of some feature of the heartbeast.
While Tytalus apprentices are no doubt often given insulting names by their master, a Tytalus mage is usually permitted to choose their own hermetic name after passing the Gauntlet. Technically it is the Tytalus Primus (not the parens) who confer the name, but most of the time that is just a formality.
Not something I've spend any time looking into in its own right, but while investigating other Latin terms for inclusion in my sagas, I have come across a few that might fit the bill:
Their existing name with a diminutive is an option - "Little Name". Unfortunately the correct diminutive to use is non-trivial (e.g. https://www.researchgate.net/publication/323534846_The_Formation_of_Latin_Diminutives_of_Nouns_and_Adjectives), but maybe the person giving the name only has Latin 4 rather than 5 anyway?
In one saga I ran, we had a Flambeau magus who had been cast out of House Tremere, and for his Hermetic name he chose the Latin equivalent of "Tremere The Founder Was A Bastard And All Of His Followers Are Bastards". He was a certamen specialist and loved antagonizing Tremeres.
In a Pendragon campaign, I named a spineless, sycophant monk NPC Brother Incipidus.
Incipidus means tasteless in latin.