Right after the section you quoted: "If a voter has not yet affiliated with a party, they are able to vote in either party’s primary election. "
So to your original point. You do not need to be affiliated to vote in the primary, but if you are affiliated you must vote in your affiliated party’s primary.
Do I have to register or affiliate with a party before I vote in the primary?
No. A registered voter is not required to pre-register or take any steps towards affiliating themselves with a party before voting in a party’s primary election. (§162.003) Additionally, when a person registers to vote in Texas, they do not register with any kind of party affiliation.
This is all on that same source that you accused the other user of not reading…