Similar to the name-change origin of the Asahina family, it appears the Oni hunted by Hiruma, Kaiu, and Kuni got the pseudonym treatment as well.
In the first chapter of Shadowlands, p. 35, it's said that the Hiruma family came about thanks to Hiruma and his followers ability in "preventing, detecting, and defeating attacks from the Shadowlands."
Well, in the original lore, it's because Hiruma, Kaiu, and Kuni were the three that said they were worthy to be Hida-no-Kami's followers, and he then tasked them with killing Hatsu Suru no Oni to prove it.
Forward to p. 53 of Shadowlands. Hiruma, along with Kaiu and Kuni, went a-huntin' as before, but... for Hajimeno Tatakai no Oni.
So, change for change's sake?