how does Britain, or other countries, maintain who can, or cannot vote ?

register to vote.

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And even worse, some do not allow you to vote in a primary unless you are a declared member of that party, and do not allow you to switch without jumping through some hoops. Yet the primaries are funded by taxes, not the parties.

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I am genuinely interested to know what the justification is for this law, because from the other side of the Atlantic, it sounds awfully sinister.

Also, it's not a monolithic practice - just over half the states have an open* or semi-closed primary, and a small number of states have caucuses. 

As to whether it's wrong - strategically or ethically?  I prefer it be a closed system, since I don't really see the benefits of an open primary - and as far as I can tell, in the states that do have open primaries have been targeted to introduce chaos into the nomination process. I'd be fine with killing primaries altogether, but I think they do serve as kind of a pre-game pep rally, and I do like feeling like I have a choice in the direction of the part I support - even though the NY primary is late enough that it's mostly a done deal by the time we get to vote. But each state has its own political character and culture.

Also, note that many states with closed primaries DO allow for unaffiliated voters - meaning that, as far as I can tell, only nine states have fully closed primaries.

I honestly don't grasp the caucus system, and I think it takes a different shape in each state that follows it. My best semi-uniformed guess is that it's kind of a holdover of the 'guys in the back room with cigars' era.

As I've said before, they failed us when it mattered most.

-C!

In Britain there is a list, called the electoral register, which contains the names and addresses of everyone who is registered to vote.

To get on the list for parliamentary elections you need to: register your name and address for the constituency in which you live (this is normally done as a matter of course when you move to a new address);  be at least 18 years old; be a British citizen or a Commonewealth citizen or a Rep of Ireland citizen; not be ineligible through 'legal incapacity', such as serving a jail sentence or being a peer in the House of Lords.

The problem that has evolved from the way primaries are done is that it has pushed the parties towards the fringes. Because only party members vote in the primaries the candidates have to please the party members, which is an increasingly small group as voters leave the parties. Also, the candidates are not voted for in order to do anything in government. Too often they lose primaries if they have done any kind of compromising to pass laws.

Primary reform is a start because it could get less extreme candidates that actually want to get something done for their constituency.