I do live in Texas and here's how the last several elections have been laid out.
Go to the court house, there is a table with 3 elderly ladies sitting behind it. I go to the first lady and show her your voter registration and she asks for your ID. I show her my ID and she asks "Republican or Democrat?" (Why that makes a difference I don't know), but I tell her Democrat (I think once I actually asked why that mattered), and she takes one of 2 little rubber stamps that either says R or D on it. And marks that next to my name and address in a big book.
This shows that I have come to vote. Then she hands my ID to the next lady and also marks a "voted" on my voter registration card and hands it back to me. The 2nd lady writes down my name in a sign-in roster and asks me to sign my name. I do that and she hands my ID to the 3rd lady who looks at it, enters some information onto a machine which spits out this little piece of paper with a 4 digit number on it. She hands that piece of paper back to me along with my ID.
Now all 3 ladies have viewed my ID and the 3rd lady has tied my ID to a 4 digit number. Maybe it's not tied together, but I find it very odd that she used my ID to enter something from it, and that generated the 4 digit code.
I then take the 4 digit code to the voting machine and enter it. It pulls up the ballot and I put in my votes. Then I review the votes and finally cast my ballot.
That's how it works where I live in Texas. In my eyes, that's not secret. That 4 digit code is tied to my votes, which is tied to my ID.