https://web.archive.org/web/20090202101557/http://cio.chance.berkeley.edu/chancellor/sp/flagship.htm
former UT President Robert M. Berdahl talks about flagships here ^^^
and Texas State is not actually a flagship for several reasons.....the main reason being that the Texas State University System is what is known as a "weak system" they do not have a flagship and all of the universities in the system lobby the legislature on their own for their funding needs instead of the system doing it for them....all of the other systems in Texas are "strong systems" with a flagship university and the system lobbies the legislature on behalf of the universities for funding needs
in addition Texas State is not the oldest university in the system and until recently they offered very few doctoral degree programs and Lamar and Sam Houston offered more of those programs and graduated more doctoral degree holders.....Texas State is by far the largest in the system though, but that is probably the least relevant criteria......in all the other "strong systems" in Texas is is pretty clear what one is the flagship without a doubt and generally that one meets all of the criteria of being the largest, the clear research university, the oldest, and the one offering most if not all of the doctoral and professional degrees
I think that TWU is back to calling themselves a "system" again they did for a number of years and then they didn't for a number of years....and I think they do again now....which is pretty dumb considering that I am 99% sure that all of their degrees are accredited under the Denton campus and nothing in Houston or dallas has it's own accreditation.....I think it depends on the ego of who is in charge there and possibly on "funding mechanisms" and getting money for "system operations" that cover admin cost that would not be covered as just a single university
in New York in the past Buffalo, Binghamton, and Albany were "cornerstone" universities in addition to Stony Brook, but in 2022 their governor shitted on Binghamton and Albany and stated that Stony Brook and Buffalo would be called flagships and all the others would "have their strengths leaned into" (which I think most people see as get fucked)
that area of the country has always been a bit weird about public education though because they have so many strong private schools that are so much older than many of their public schools that they have let their public schools kind of plod along and be looked at as decent "back up schools" or "well Rutgers it is" (or UMass, UConn, SUNY-(whichever) etc.)