Lots of names in Texas.
I live on the edge of Houston. Houston is known as Southeast Texas, the Gulf Coast, Greater Houston, or simply Houston. Houston could mean anything from Conroe to Baytown to Clear Lake to Pearland to Richmond to Katy to Cypress.
I moved here from Fort Worth. That part of Texas has lots of names. It's known as the Metroplex, a chamber of commerce creation from the 70's that stuck. It's known as Dallas-Fort Worth, or DFW. In the last 10-15 years, as the area has expanded from two primary counties to four (Dallas, Tarrant, Denton, Collin), it has become known as North Texas. Outsiders simply call it Dallas. (those are fighting words in Fort Worth)
West Texas starts anywhere from Abilene to Weatherford, depending on who you ask. I'd call it anywhere from where you can't commute into the DFW area.
East Texas is really anything north of Lufkin (maybe even Livingston) and east of roughly TX 19. Anywhere from where you can't commute south into Houston or west into DFW. The Lufkin area is also called Deep East Texas or the Piney Woods. Yes, in Houston, we say "up in East Texas", even though we're also east.
Anything south of Austin is South Texas.
Waco, Temple/Killeen, and Austin can all be considered Central Texas.
Bryan/College Station is considered Aggieland. Or, these days, College Station. Bryan was traditionally the larger city and is the county seat, but College Station has outgrown it with the growth of Texas A&M.
Beaumont/Port Arthur/Orange has always been called the Golden Triangle. It is also considered Southeast Texas, both separate from and combined with the greater Houston area.
The Corpus Christi area is known as the Coastal Bend. Or, simply Corpus.
The Rio Grande Valley is used for the three counties at the bottom tip of Texas. It is shortened to "The Valley". I've never been to the Valley.