In french, a caret is used to mark a "lost" character. It was taught to me as a grave stone. Hostel -> hôtel, which became hotel in english, and hostel became a different thing.
Note that the caret (‸) and circumflex (^) are different signs. The caret is used across plenty Latin orthographies to convey that something is missing, as its name implies; while the circumflex is a diacritic usually going over the letters, whose function depends on the orthography of the language in question.
EDIT: as for the etymology of circumflex it's basically Ben Dover "bent around" (circumflexum).
Fun fact, in french, the circumflex usually means that there used to be an S after that letter, but it was linguistically evolved away over time. Depending on when this happened, we can see remnants of the before version of this evolution in some English words
Yup, in French that circumflex is kind of etymological. I say "kind of" because that /s/ being dropped changed the pronunciation of the preceding vowel, and depending on the vowel and the modern dialect there might be some leftover of that change; for example ⟨tâche⟩ /tɑʃ/ "task" vs. ⟨tache⟩ /taʃ/ "stain".
Originally the diacritic backtracks all the way into Ancient Greek. Back then Greek had a pitch accent, and a vowel could either raise in pitch (so it got an acute, ά) or fall (so it got a grave, ὰ). But some long vowels and diphthongs did both things, raising then fall, so the solution was to mark it with both, as ᾶ. Eventually that circumflex evolved into a tilde-like shape, but that's a coincidence.
Other languages might use it for vowel length, vowel quality, stress.
There's also the upside-down circumflex used in Chinese transliteration which indicates the opposite kind of intonation change. 我 (wǒ) has tone which falls and then rises (like a person confusedly saying "huh?")
It depends on the dialect, as it relies on the weak vowel merge. Basically: if "John Lenin" and "Vladimir Lennon" sound right for you, you got that merge.
For those who find them to sound the same, the second vowel should be around [ə]. For those who distinguish them, "caret" should have [ɪ] and "carrot" [ə].