Open primary. Texas doesn't have party registration, which is nice because party registration is silly and it shouldn't be the state government's job to ensure party loyalty or the ideological purity of primary voters.
Well put. If the state is paying for the primary, I don't see why the state should try to enforce the desires of a party to have only certain voters. If the party wants to pay, then they can set up a pre-registration system for their party. IL uses an open party primary for almost all races, but township party organizations may hold closed caucuses to nominate their slates for township officers.
I thought this was a loophole, where "new" parties could nominate by caucus; so each election a new party would qualify, and choose a name that had the same initialism (eg League Of Loggers, Laughing Out Loud; Loophole Of Lincolnland; would always be "LOL")