To add to this: classical extemporization, post-Romantic period, also sort of fell out of favor as the composer began to pre-empt the performer and the "written note is sacrosanct" effectively became the de jure standard.

There's a great story around how Beethoven, perhaps one of the strongest improvisers of his day, completely upstaged Steibelt, a contemporary musician and by all accounts a bit of a charlatan. I'll include the entire quote verbatim from "The Lives of the Great Pianists" by Harold Schonberg.

  This time Steibelt had prepared a brilliant fantasy for piano and strings, the theme coming from the Beethoven trio he had heard the previous week. Steibelt's admirers were in raptures. Now the issue was joined, and Beethoven had to show his strength. He walked to the piano, grasped the cello part of the Steibelt work en route, put it upside down on the piano and insultingly drummed out a theme with one finger. Then he improvised, and—angry, excited, on his mettle—how he must have improvised! Before Beethoven had finished, Steibelt stole from the room. He never again would meet Beethoven and made it a condition before going anywhere in Vienna that Beethoven not be invited.

I'm a little lost. What am I listening for in that video?