I'm a Boomer, and when I tried to find some GenX music that I could say I liked, more often than not the musicians turned out to be Boomers that I had discovered late, or Boomers who had bloomed late. It's not that I don't like music by musicians born between 1965 and 1980, it's that my stereotype of the period was that it's kind of tweener--hearkening back to Boomer music (like REM, who are actually late blooming Boomers) or anticipating the 90's (depending on how you demarcate grunge).

But my stereotype seemed too confining, so I looked at the chronology of the music I listened to most frequently on Amazon and discovered the following musicians as having been born Gen X and being so distinctive that they were not "tween" anything:

Bjork

Jeff Tweedy

Erlend Oye (Kings of Convenience)

Montt Mardie (David Pagmar)

Andreas Mattson

Annika Norlin (Hello Saferide, Sakert!)

Martin Molin (Detektivbyran, Wintergatan)

Loney Dear

Irene Tremblay (Aroah)

World’s End Girlfriend

Jeff Mangum (Neutral Milk Hotel)

Godspeed You Black Emperor

Elliot Smith

Rose Melberg

Ben Gibbard (Death Cab for Cutie)

Obviously I tend to favor the outliers. But if you add these musicians to those that you and Keith listed, it's fairly obvious that Gen X musicians should not be stereotyped or relegated to a "between" category. They are uniquely unique. I had not thought of the members of my list as Gen X, but they're clearly not Boomers or Y / Z / Millennials. There's a real distinctiveness to artists from your era, and we may be just starting to understand how complex and rewarding their artistry is.

P.S. from my own list, my personal favorite is Annika Norlin. As much as or more than the others there, she has a Boomer sense of history with a post-Boomer sophistication / irony--imo the most interesting musician of the present century.