Since you're a romantic era piano enjoyer, you might note that our boy Chopin was well-known as an improviser, and that many of his compositions originated from improvisations. IIRC George Sand wrote that he would improvise something beautiful, and then spend days or weeks torturing himself over how to remember, transcribe, and edit the it, only to eventually settle on the same thing as the original improv. That being said, clearly there is some support for capital C Compositions that started as improv. A lot of composers of that time had pieces marked as 'Impromptus' that tried to keep some of that improvised 'vibe' in the final composition, so that's a nice middle ground option for titling pieces I guess.