Your challenge is to write a program that is a quine, meaning it prints out its own source code.
For at least one individual character, when it is removed from the program's source, a single different character is removed from the output.
By different, I mean that the character removed from the code is distinct from the character removed from the output - not just in a different place, but a different character altogether.
Your program, modified or unmodified, may output to stderr as long as it also outputs the correct text to stdout.
Your score is the number of character this doesn't work for, with lower being better.
For example, if the program
abc is a quine in your language, and the program
ab, your score is 2, because removing one of the chars works and removing the other two doesn't.
I have a 200-point bounty for anyone who gets score = 0. This may be increased, depending on how hard it appears to be. Bounty has been awarded to Jo King's answer.
I updated the criterion to reflect the fact that scores can converge to 1 by adding a large amount of characters.