I'd like to add a bit to some of the above. That is relating to the question of authenticity, or adherence to "what the composer really meant" -- sort of hinted at in avanchnzel's answer.
It is quite true that most urtext editions have been heavily researched and corroborated by looking at as many manuscripts as the publishers can find. This does not mean, however, that they represent what the composer really wanted. Sometimes, no doubt, they do. Perhaps even most of the time. But sometimes -- particularly with baroque and earlier, including Bach -- they don't, or, more accurately, they become that group of scholars' opinion as to what the composer really meant. Neither they nor anyone else has any way of knowing whether they are correct.
To take Bach as an example. For some works -- the Well Tempered Clavier, or The Art of the Fugue, for examples, you are looking at an academic study, and it is possible to be pretty sure of what dear old Johann meant. With the organ works, however -- all of them -- the man was a master improviser, and it is highly unlikely that he ever played them quite the same way twice (even when he wasn't borrowing from Cantata A for a fugue in some other organ piece).
So one can quite legitimately have differences of opinion as to which particular set of notes was really meant.
Also true of modern French organ music -- the whole crew; Widor, Langlais, Alain, Dupre, Franck, etc. -- they were improvisations before they were ever written down.
Take the pontificating of scholars with some caution.
Extremely well put, the accuracy of your statements is to be highly complimented. And, for those naysayers in the UK who disagree with my thesis of original performance, I state the following:
1) The leading applied musicologist on this subject (in the world, much less the UK) is Dr. Clive Brown of the University of Leeds. He stated in an interview (March of 2015) with radio station WQXR of New York city the following: "All Classical Music is Being Played Wrong."
2) As accurately stated in this particular post, he said that all musicians of the time commonly improvised and rarely played anything the same way twice (Mozart and Beethoven being the most well known examples).
3) Guess who has completely endorsed the thesis of my video on several occasions. Oh, my!
4) Finally, the term Urtext was originally coined and espoused by Heinrich Schenker in the late 19th and early 20th centuries. His work was mostly concerned with the late Beethoven Sonatas, with which many performers were taking great liberties.
5) Not only did they improvise (like Beethoven), they were also publishing printed versions of their performances, which in a word was: wrong.
6) However, for Schenker to haul out Beethoven's manuscripts and state that this is the way the composer (the great improviser) played them is total falsehood. The man was totally deaf and had not performed in years.
7) During his lifetime, Schenker had to teach out of his home because no one in any music conservatory would give him a job. Why? Because no one, and I mean no one played anything literally, according to the score.
Cool Then, as accurately stated in his Memoir, Earl Wild reveals that directly after World War II Adele Marcus, Rossina Lhevinne, Claudio Arrau, Rubinstein, et al, resurrected this note perfect, attention to the score Urtext methodology.
9) If it take us the rest of our lives, Clive Brown, Neal Peres Da Costa, Kenneth Hamilton and myself will let the rest of the world know how the composer/pianists who originally wrote this great music actually performed it!
Topic: Is it important to get Urtext ?