DAs and DAAs are different.
The former are just humans that have become quite explicitly inhuman via magic. Usually magus attempting to achieve immortality or progress their Root research via augmenting their body/spirit. Typical abilities include being able to feed on the spirit/life force of humans and turn the empty husks into their ghouls.
Sometimes, a person fed on by a TA/DAA/DA will become a DA
DAAs are a rank. The originals were the first and strongest DAs created directly by the True Ancestors and kept by them, or similar equivalents. Primate Murder, for example, is mentioned in Tsukihime as a DAA and is a creature created by Gaia to kill humans, but considered a DAA because it's controlled by a TA. The relationship to TAs, and by extension to the Crimson Moon, is the defining factor of DAAs.
And True Ancestors are beings created by the Crimson Moon to be its avatars, to be autonomous anti-humanity weapons, and to serve as Gaia's general purpose defense force. They are the Crimson Moon's equivalent to Fate's Divine Spirits. Their common trait is an instinctive bloodlust that drives them to prey on humans (despite not really needing to in order to survive) and the ability to turn human victims into their slaves, either by rendering them mindless ghouls or by creating DA's
In Tsukihime, the TAs have been all but wiped out and the DAAs basically run amok free of any kind of opposition or control. Only the church can oppose them. They're the #1 antagonist. But, theyr existence depends on the presence of the Crimson Moon, and the Crimson Moon is only around in Tsukihime because Gaia considered humankind a threat and offered it a contract to cull them. In Fate, Gaia supports humanity, which is why the Throne of Heroes exists at all, so both servants and the Crimson Moon can't share the same setting, meaning TAs and DAAs can't coexist with servants.
For the record, don't read the wiki. It's full of hearsay and straight up fanon garbage.