I think the issue here is more one of America's political institutions than of language. So let me explain in terms of the politics and let the language fall out from that.
In the US, the way we elect our president is, basically: The voters choose who they want for president. But we don't simply count the total number of votes for each candidate. Rather, each state gets a certain number of votes depending on its population. Then all of that states' votes go to whichever candidate got the most votes in that state. So for example, my state of Michigan gets 16 electoral votes. There are about 5,400,000 voters in Michigan. So if 2,700,001 voted for Jones and 2,699,999 voted for Smith, Jones would get all 16 of Michigan's votes.
(How elections are conducted in each state is up to state law, so a few states, Nebraska, Maine, maybe there's another one or two I'm forgetting, will split their votes.)
Mechanically, the way this is done is that when the citizens vote, they are not REALLY voting for a candidate for president, but rather for an "elector" who is committed to voting for that candidate. So as Michigan gets 16 votes, Michigan will choose 16 "electors". Each party puts up a slate of electors. So the Republicans nominate 16 electors, the Democrats nominate 16 electors, and any other parties who qualify to be on the ballot each nominate 16 electors. Whichever slate wins, their electors get to vote.
The electors then meet at the state capital a few weeks after the popular vote to officially cast their votes. These votes are then delivered to the national capitol where they are officially counted in front of the Senate.
Some states have laws that say that an elector is required to vote for the candidate he is pledged to. Other states say that, theoretically at least, an elector could vote for anyone. And every election we have there are always a handful of electors who vote for someone else. These are called "faithless electors". There has never been a case in US history where electors changing their vote has changed the outcome of an election.
In the 2016 election, when Mr Trump won, there was a concerted effort by Democrats to convince Republican electors to vote for a "compromise candidate". The plan failed. Only 2 Republicans switched their votes: One voted for John Kasich and another for Ron Paul. Meanwhile 8 Democrats switched their votes: 3 for Bernie Sanders, 3 for Colin Powell, 1 for John Kasich, and 1 for Faith Spotted Eagle. (In my humble opinion, the plan was pretty far-fetched. When the outcome is in doubt, someone might agree to a compromise. But why would you agree to a compromise when by doing nothing you win?)
So all that said, what the text you quote is saying is that in some states, they show the name of the electors on the ballot along with the name of the presidential candidate they are pledge to vote for. In most states, they just list the presidential candidate. So in most states, most voters don't even know the name of the elector that they're voting for. Because it really doesn't matter. I saw one news story about the electors in my state meeting to vote and the reporter described the electors as "getting a footnote in history".