I just found out that each precinct may use different methods. Our Republican ballots do not have names on them, though. They are blank pieces of paper. So I can vote for anyone I want. I live in a ‘village’ so only 25 people will show up - that is probably why we sit around Democrat-style and try to change each others’ minds. Also, in some other smaller precincts, a shove of hands may be used. But, you are correct that, in general, Democrats openly talk to each other and try to get supporters to join their group, while Republicans vote by secret ballot.
I’ll be damned. I have heard rumors of that but thought it was BS. There are no set rules as far as I know other than report the totals to the state whey you’re done. It’s pretty laid back.