Perhaps the references to East, West, North etc. Texas have to do with the fact that when Texas was admitted to the Union, the enabling act allows Texas at any time to be divided into as many as five states. An analogy is West Virginia which was part of Virginia until the Civil War, when it broke away and became a state of its own. Its as if East Texas, etc. are states that could be born.
Contrast with Northern and Southern California or Southeast Wisconsin. These areas were never thought of as states of their own.
New Jersey, for one, doesn't observe your rational rule. As New Jersey is a fairly narrow vertical state, its areas are commonly referenced as North Jersey and South Jersey (and Central Jersey). As far as I know, there are no legal divisions, or possibilities of separate statehood.
In official documents, however, I would expect to see Northern New Jersey etc.
Maybe it has something to do with syllable count and euphony.
The State of Washington is usually referenced in local news stories by NW, SW, NE, SE, Central (Yakima, Ellensburg). The Cascade Range divides us West & East.
We use North Georgia for most of Ga starting at Atlanta. Almost never do we say South GA or Central GA. The mountains are almost always said be be North Georgia Mountains.
I don't think there's anything special going on in Texas. I grew up in Pennsylvania. Depending upon where you are, you are in NW, SW, N central, S central, NE or SE Pennsylvania. They are common regional terms. I now live in S central Wisconsin. This state is also spoken of as various regions.