"Because they were always "the band" to various frontmen and the locals in Woodstock ..."
https://en.wikipedia.org/wiki/The_Band
:)
They were originally called the Jayhawks(correction: the Hawks) and ran across another US band by the same name...so they was going to change it to "the Jayhawks or Jayhawkers , but didn't like that name....this was iirc, after Dylan had his motorcycle wreck and moved in up northern New York state where Dylan and the Band were at the Pink house doing the basement sessions....the Band had started backing Dylan right before all this took place and instead of thanking them by their band name he would say "give it up for the band"....so they kept it as their band name during those basement tape sessions
TC
BTW, I haven't bothered to check your Wikipedia link, because I watched Robbie Robertson and Levon Helm describe it just as I've posted many years ago