Re: Most countries record the names of voters without making that information "public"
In France, every time I vote I go through the same procedure. I bring my voting card (sent regularly by the government), present my voting card to the first town official who looks me up in his database - which is a sheaf of printed papers with all the people allowed to vote in that town. When he finds me, he has me sign in (with a pen) in the empty space next to my name.
Then the second town official gives me my voting envelope (which is empty) and I can go to the table where the ballots of all available candidats are waiting for me to pick up (one ballot per candidate). I look over the choices, pick more than I intend to use (because I protect my voting anonymity that way), and I go into the voting booth to place my chosen ballot in my envelope. I could also have brought my ballot with me, because a week or so before the vote I am always sent an official, nominative envelope with all the candidate's ballots in it.
When I exit the booth, I go to the voting box where a third town official verifies that I have placed my envelope in the box, saying "a voté !" (has voted) when I'm done.
Then I go to the fourth and last official who gives me back my voting card and I am free to leave.
So my name and signature are properly registered on a piece of paper which is undoubtedly stored somewhere, but who I voted for will never be known unless I say it.