Those are three excellent descriptions of what Habeus Corpus means, historically, and both the writ (issued by a court) and a petition (made on behalf of someone to a court, soliciting release). The campaign to "restore" habeus corpus actually contradicts HarryVan's statement that, "It has never been abrogated in the U.S. as it has in otheer countries."
The idea behind the "restore" campaign is that true habeus corpus was suspended by the (socalled) Patriot Act. The act provides for imprisonment of enemies of the state (generally these days they are called "terrorists", but the definition is VERY broad and includes people who "give aid" or support to terrorists, or organization believed to support terrorists). The point is that no proof of wrong-doing is required to place these people in a jail (the special jail at Guantanamo, where we also have a Navy base on the island of Cuba, is especially controversial because it was created specifically to avoid legal issues like habeus corpus, and the burden of proof principle). All that is required under the patriot act is the accusation that the person is, or even just MIGHT be a terrorist, and the government is allowed to lock them up with no time limit on a)When they must be released, or even b) when they must be brought to trial to determine guilt or innocence of any crime or wrong-doing.
The "right" to a speedy trial is in the US constituion, as is a provision that the punishment should be in proportion to the crime ("cruel or unusual" punishment is not justice is the way that is generally stated) I believe (though I am often wrong about many things).
So essentially the campaign to bring back the right of habeus corpus is a campaign to correct the injustice(s) of the Patriot Act that attempts to throw out a person's right to know the crime he/she is being accused of, of the right to speedy trial, and several other things including a "right" to reasonable privacy, which includes both a right not to have police search or "sieze" (confiscate) any property without prior authorization, and a reasonable expectation that one could hold a "private" conversation (as in a phone call, or email) without the goverment being allowed to look at or listen in to it.
Another recent act of Congress allowed for the National Security Administration to "intercept" communications INSIDE the US (previously they were only allowed to listen in on communications OUTSIDE the US) as a kind of "extension" of the Patriot Act's idea that we need to watch terrorists wherever they are. This came about partly because internet communications and much of the world's telephone traffic now travels THROUGH the US even if it is on its way from (for example) Germany to Saudi Arabia. It just has become more "practical" for the NSA to "listen in" on this foreign traffic as it passes rather than go out and seek a method of listening in both Germany and Saudi Arabia (which they were doing before anyway). But this other "act" goes further to legitimize the (previously "illegal") tactic of the current Bush Administration of wiretapping conversations between Americans in the US and other persons (citizens or not) outside the US. (Which the NSA was doing anyway even before the Bush administration, but they would have to do it FROM Germany or Saudi Arabia or some other foreign place.)
Generally Americans don't want to give the government the "right" to listen to their private conversations without going through the proper channels (the courts) to seek permission, and having to at least show the court some evidence (not necessarily "proof" but some evidence) that listening in on those people is in the interest of justice and the public good.
Since these "rights" are not only believed by most people to be "natural rights" that belong to everyone, but in our case are also GUARANTEED to us in the US Constitution, it seems to many of us that these things should never have been allowed to be provisions of a less important law, and are therefore "unconstitutional" (and therefore unenforceable). There is a famous old saying (no, Ben Franklin did NOT say it) that goes roughly, "Those who give up fundamental rights to secure a little temporary security, deserve neither their rights nor can they expect security."
I hope this has helped you understand this more fully.
Sincerely,
Stafford "Doc" Williamson