IIRC, it was named that for being a patchy system, though?

Yes, it was called A patchy server initially, and then that was made into a wordplay on the term Apache, and they used a colourful feather as a logo.

There is also sn interesting anecdote about Apache devs, IBM and open source.

Has the Apache Nation had a dispute with the ASF? Or just some activism group?

Not that it matters, really. The all-girls' Catholic school, Ursuline Academy of New Orleans, has a sorority system based on year of graduation. The classes/sororities were called Skips (Skipperettes), Macs (Merry Macs), Sioux, and Leps (Leprechauns). The school sought, and received, permission from the Sioux Nation to use the Sioux name. Despite this, after Black Lives Matter happened there was a nationwide reexamination of whiteness's impact on our culture, particularly with regard to cultural appropriation, and the decision was made to change the name of the Sioux class to Phoenixes (or Nix for short). So the Apache Software Foundation probably won't be named such for much longer.