I've always been of the opinion that Tex was a completely separate smart A.I., like Alpha was before he was tortured. However, recently I noticed something that Delta says in Chapter 7 that caught my attention.
Delta: There is one smart A.I. in the Freelancer program.
Delta: The Alpha is a fully formed-
Wash: Delta! That's enough.
Delta specifically says that there is only one smart A.I. in the Freelancer program. And specifically says that one is the Alpha, implying that Tex isn't a smart A.I. After spending a small amount of time with Church, Delta was able to figure out that Church was the Alpha without a lot of apparent evidence.
Wash: Delta was the logic. He was able to figure out things before anyone else. It's why he left a message for you in a way that he knew only you could find, and in a way that let me see you getting it.
Delta was obviously way ahead of everyone on this. And if he could figure out Church so easily, he certainly could've figured out Tex, who he was more acquainted/familiar with. But he obviously didn't suspect anything. Again, I believe that this indicates that Tex isn't a smart A.I.
Which leads me to two options - either an iFrag (fragment of the Alpha) or another type of A.I. altogether. If she is an iFrag, it seems to be likely that she would be a larger or earlier portion - in this case, I'd guess Beta. Her actual "personality trait" could be a number of things, or perhaps she was some part of the Alpha that was still attached to Allison, in love with her, and grieving for her.
I'm not sold on Tex being an iFrag - she seems too complete and too important within PF (Project Freelancer) to just be another of the Alpha's fragments. Additionally, Wash has abosolutely no clue what Tex is - "I don't know what she is," and "Tex is... confusing." Epsilon would've been the last A.I. to split off (which is why Wash knows all the things he knows - Epsilon would've gotten memories of all the torture that happened before he split). If Tex were an iFrag, Epsilon would likely have had memories of Tex and her formation. If this were true, Wash would've seen those memories, some or all, once Epsilon started losing control, giving him knowledge about Tex.
So this is my theory: Project Freelancer attempted to make another A.I. after the Director's memories of Allison, for two reasons: (1)they could only receive one A.I. for their experiments, and (2)the Director was haunted by her memory, and attempted to do something about it by making an A.I. of her. But since it wasn't actually based on her mind, it couldn't be created as a fully-formed, smart A.I., like the Alpha. Rather, what they got from it was some kind of lesser A.I. So, Tex would be neither an iFrag nor a smart A.I., but is a stand-alone, lesser A.I. Still powerful, but different. (Which could also be one of the reasons why she worked so closely with the Director - not only would Tex be inclined to be the Director's favorite, just by the nature of their relationship, but they could've also worked closely to attempt to augment her capabilities as an A.I.)