There are really three different systems in use in the U.S., and which system is used differs from state to state.

A caucus is where there are local meetings for each party where delegates to a state convention are selected, based upon the candidate(s) chosen at the meeting. Registration is required and voting is public (i.e., if you vote for a particular delegate/candidate everyone at the meeting potentially knows how you voted.) Registration is necessary to ensure every individual attends only the meeting for the party of registration. The actual candidates are chosen at the convention.

A closed primary is an election, where your vote is private. You must be registered to vote, including naming your party of choice (your party of record,) and you are supplied a ballot listing only that party's candidates. You may not vote for a candidate not belonging to your party of record.

An open primary is also an election, where your vote is private. However, while you must be registered to vote you need not declare a party of record (and, in most cases, the question is never asked.) Anyone may vote for any candidate from any party.

Personally, I have never participated in a caucus or a closed primary, only an open primary.

That's not the way it is in Wisconsin.

In Michigan you do not have to declare a political party to vote in the primary. Though you can only vote in one, not both.

Some say it's because it would leave the door open for everyone who has an opposing ideology to vote for who they think is the weakest candidate so that their real candidate would have an easier chance at winning when they face the weak candidate.

I think it leaves out Independents who are usually the deciding factor. I'm a registered Democrat. I would be a registered Independent (even though I lean more Green) if NY didn't have a closed primary. I hate these labels. As soon as you say you lean Green, people think you're a radical leftist. People need to learn these definitions instead of listening to idiots who think everyone left of them is Karl Marx.

I like your idea. I would change it to voters can vote for one person in the primary, doesn’t matter which party. Then the top 3 or 4 primary winners would head to the general election.
Term limits would be good, as would getting the money out of the elections, maybe forcing candidates to pass a civics test.

Being able to vote at all in primaries is a rather recent development in itself. The Dems decided their candidate without a public vote all the way up until 1972. The Republicans were slightly earlier having started a popular vote in 1956.

Primaries differ from state to state. Some states have "open primaries " where you are free to vote for anyone from any party. Other states have " closed primaries " where you can ONLY vote for a candidate of the party you are registered with. Either way, our democracy is completely messed up and is actually an oligarchy. I hope this helps clear it up a little.

We need these things:

1. Publicly funded elections (elimination of PACs/dark money)
2. Elimination of super delegates in Democratic Party
3. One person, one vote (no more electoral college)

If it was left to me, money would not be an issue for who can run and what they can do to get elected. That would open it to everyone and hopefully get a better representation of the people.

Well, if you get right down to it, we do it in a lot of things. An example of this is Football. We have divisional races, the conference and finally afc/nfc play for the world championship. Okay dumb example. But it’s a system that is not just limited to political parties.

Might enables "gerrymandering" to be optimized.

It's to give the political parties more power. By splitting the voter bases.

I think this is a state by state thing. I'm not a member of a party and regularly vote in primaries. Only can vote in one though.

The original idea was so you couldn't go and vote for the opposing party's worst candidate...but that's what we ended up with, anyhow! Florida is considering open primaries. there are a few states which have them.

The parties are private clubs. They are in no way legal authorities until they are elected to an actual position. Then the person affiliated with that party has the power, not the club.