Latin was also the lingua franca of nobles and merchants. Lingua Franca Mediterrania itself, the language spoken by medieval sailors in Italy, Egypt, and nearly all the coastline of Mediterranean is a creole of Italian, Portuguese, and Arabic. To the point that Shakespearean Moors spoke it as "Arabic".

Then, the language of science until the 1900s is Latin. Isaac Newton wrote in Latin. Nearly all scientific papers and books were published in "Neolatin".

Latin was also the language of academia, which gradually separated from the Church starting in the Late Middle Ages.

Both. (Less and less towards the end, of course).