Maybe its only my opinion, but I risk to write it anyway:
Daito Ryu Aikibudo is a mainline Daitoryu, after re-arrangements of techniques into present groups (such as Hiden Mokuroku Ikkajo to Gokajo) by Takeda Tokimune Soke. However, Takumakai also cooperated, and has similar syllabus, just the way of performing some waza are different.
So simply saying, Daito Ryu Aikibudo is a Daito Ryu Aikijujutsu as taught by Takeda T. Soke. Kondo Shihan was a student of late Soke, and what he teaches is Daito Ryu Aikibudo of Takeda T. The only question we may ask, why the name Daito Ryu Aikibudo?
I may guess, that its done to represent the mainline
Daitoryu, descending from Takeda family. Other groups, like Kodokai and Sagawa Dojo didn't follow same path as Takumakai, and their curriculum/scrolls are different. Every group emphasizes something special, unique to itself. Kondo Katsuyuki Somucho teaches the same techniques and in a same way as Takeda Tokimune Soke did.
I know its confusing, but that's how it makes logic to me. So at the end, both names are correct and righful to use by Kondo Somucho.
Paul Wollos