The Jedi totally would have had their own language. They are a somewhat insular community that has existed for 25,000 years. Yes, they're taught Basic and a bunch of other languages, but in the Temple? They've got their own linguistic thing going on.
Counterpoint: it's less their own language and more that having to learn a bunch of languages to be good mediators and ambassadors, "Temple Speak" is just a wide-spanning creole that has bits of dozens of languages

















