Many of the arguments linking Texas to the "Deep South" hinge on our involvement in the Civil War. There's no denying Texas's alliance with the South in that conflict, this state was a member of the Confederacy, and that's that. To some people, that forever brands Texas as part of the South, which is a fair argument to make. The people who I most often see using Texas's alignment with the South during the Civil War as irrefutable evidence that we're part of the South tend to be folks who personally have strong connections to the culture and traditions of other southern states. And it's true, Texas was part of the Confederacy and a slave state, so those two sad facts make a strong argument for why Texas is a part of the South. My position is not an apologist attempt at ignoring shameful connections to slavery or denying that Texas fought for the Confederacy, but simply to say that modern day Texas is too different from true Southern states to be grouped with them now.
I believe this to be true because of vast demographic and cultural changes that have occurred in Texas since the end of the Civil War. Traveling in the true Southern states, it's impossible to ignore the still strong cultural identity that many Southerners seem to feel in connection with the Antebellum South. Most modern Texans I know couldn't name a Civil War battle fought in this state, but they sure as heck remember the Alamo and Battle of San Jacinto. In the rest of the South, connections to things like plantation life and slavery just don't seem that far in the past, there are still deep cultural ties to those things, and pride in being part of the Confederacy. Here? I'd argue that more Texans care about our state's battle to win independence from Mexico than they do about the Civil War. That conflict left lasting scars in Texas, but it is not the defining moment in our history that it seems to be in a lot of The South.
As time has passed, it seems to me that most of this state has slowly cast off remaining ties to a truly Southern past. There are high schools and streets that still retain the names of Civil War generals, but fewer and fewer seem to remain. Texas was always, at most, an outlying state in the South, and large enough to encompass many different cultures, something most of the rest of the South can't really claim. I doubt that many people would notice much in the way of true Deep Southern culture in cities like Austin, San Antonio, El Paso, or even modern day Houston. While we're also part of the Bible Belt, Texas has always had a larger Catholic population than other Southern states to moderate the "WASPiness" of the culture and keep organizations like the Klan from getting as strong a foothold as they once had in the Deep South.