I’ve heard “moot” used two different and, apparently, contradictory ways –
A point of dispute that has become irrelevant to the topic at hand.
A point of dispute for which there are arguments on both sides and which is appropriate for current discussion.
Has anyone else noticed this. What do you mean when you say “moot”?
I think it means this post belongs in another forum. As to the OP, it’s a moot point.
(That’d be closest to #1)
From Rick Springfield’s “Jessie’s Girl”
‘…I wanna tell her that I love her but the point is probably moot…cause she’s watchin’ him with those ayes, and she’s lovin’ him with that body - I just know it, and he’s holding her in his arms late late at night…
ya know I wish that I had Jessie’s girl…’
(All from memory)
In Rick’s case, he uses moot as a meaningless point: he wants her to know he loves her, but she loves someone else, so…what’s the point? He made a point about telling his love for her…but , pretty useless when she doesn’t love him.
The point is moot. Kinda fits your number 1.
The original usage, and that given by the OED, is #2; it’s the meaning that is most true to the etymology, and the best synonym is “arguable.”
Meaning #1 is more common in U.S. usage, however, to the extent that Bryan Garner, normally a prescriptivist, says in his A Dictionary of Modern American Usage that “To use moot in the sense “open to argument” in modern AmE is to create an ambiguity and to confuse readers.”
The question is moot! I get the car.
I figure a point is moot if it neither helps nor inhibits the discussion at hand, i.e., it’s irrelevant.
I always understood it to mean that any possible conclusion to a question being raised would have the same impact (usually none at all, but not always) on the issue that caused the question to be raised in the first place.
Example: Ad firm is asked to represent a client that makes greasy widgets. Ad firm already makes ads for a company that makes greasy grommets. An associate asks whether or not there’s a conflict of interest. Debate on whether or not widgets and grommets are competing products in a greasy world. In mid-debate, word comes through that the greasy grommet company has jumped ship to another ad agency. Argument is now a moot point: competing products or not, the ad agency may proceed to represent the widget company.
Example II: The high theocrat states that all atheists are unGodly by definition, and that cannibals are unGodly through their sinful dietary habit. The village theocratic emmisary says John should be stoned to death for cannibalism because of this, since John doesn’t attend church and is therefore obviously an atheist. Local village elders argue about whether or not John is indeed an atheist or simply a non-churchgoer. Argument is and has always been a moot point since the question of whether or not John is an atheist does not address the question of whether or not John is a cannibal, even if the high theocrat’s words are gospel truth.
As explained by The Word Detective, this all has to do with law students and moot court. The original meaning of the word “moot” was “a point that could be argued” (the OP’s meaning #2). So when law students practiced by arguing real, disputable cases, it was called “moot court”. But because the students were arguing cases that had already been decided elsewhere, the students’ decisions had no relevance (the OP’s meaning #1). So the word “moot” is gradually sliding from meaning #2 to meaning #1, and it’s all the lawyers’ fault. 
That may as be, but if it was a gradual slide, it’s been pretty well slid for a long time now. When used as an adjective (at least in the U.S.), moot means a point or question that has no bearing on the central issue you’re discussing. However, I’ve heard the verb used both ways – to moot a question by making it irrelevant through other factors changing lessening its importance, or to argue a point (typically “to moot it about” or “moot it out,” etc.).
–Cliffy
I’ve never heard or read “moot” as a verb. My own usage conforms mostly to definitiion #1, with the caveat that the point being discussed has not become irrelevant (in the sense that the discussion drifted to matters irrelevant to the topic) as much as the topic has moved on to other things. It’s like debating the merits of competing brands of buggy whips. The debate is moot, since the need for buggy whips has effectively dropped to zero.
Another related possibility is that the discussion has become bogged down with minutiae and trivial points. Which buggy whip has more “crack” in it? Who cares? It’s moot.