If you want to get to the heart of the truly dysfunctional relationship between Yahoo and Microsoft, consider the alleged 24-hour deadline that Yahoo claimed Microsoft and its loyal sidekick, activist investor Carl Icahn, gave the company to respond to the pair’s most recent joint proposal to settle their differences.

I say “alleged,” because as in all things related to this takeover mess, the trio disagrees on exactly what even that meant.

Frankly, it’s enough to make one think former President Bill Clinton’s definition of what “is” is makes more sense.

Yahoo (YHOO), in a rather strong statement Saturday night, said Microsoft tried to jam the company into swallowing a semi-sweetened new search deal with a side order of Icahn control.

But Microsoft (MSFT) sources scoff at the notion, noting that they only wanted to try to move a deal forward more quickly than the previous interminable roundelays that have exhausted everyone.

Before we get to this latest disagreement, here are the terms of the new Microsoft/Icahn joint deal to take control of Yahoo’s search business and hand over the rest to Icahn, which, according to numerous sources from both sides, was quickly rejected by Yahoo Saturday.

The deal included:

– $1 billion for Yahoo’s search business and a five-year guarantee of $2.3 billion in search ad revenue, with an option to renew it for another five years at a $1.6 billion minimum;

– An offer by Microsoft to buy $3.9 billion of Yahoo shares, and lend the company $2.8 billion at a five percent interest rate, by taking over a part of its debt. The money would be used to give a special dividend to shareholders;

– An agreement to raise the TAC rate (a payout to Yahoo on each search query) to 85 percent from its former offer of 70 percent, for three years, and to 75 percent after that;

– A plan, unclear as to specifics, to spin off Yahoo’s Asian assets, with money going to shareholders;

– And, last of all and the obvious dealbreaker, Icahn would get control of the rest of the company, which includes the massive content and communications assets. Apparently, one or two current Yahoo board members could possibly stay on. Presumably, Icahn could then strike a deal to merge those with assets of News Corp.’s (NWS) MySpace or Time Warner’s (TWX) AOL. (News Corp. is the owner of Dow Jones and of this site.)

But, Yahoo rejected the proposal in strong terms, especially stressing that it would not negotiate such a big deal under a time constraint.

Here’s Yahoo’s official version, from a statement :

“The proposal was made on Friday evening and Yahoo! was given less than 24 hours to accept the proposal, the fundamental terms of which Microsoft and Mr. Icahn made clear they were unwilling to negotiate.”

And Chairman Roy Bostock piled on: “After negotiating among themselves without the involvement of Yahoo!, Carl Icahn and Microsoft presented us with a ‘take it or leave it’ proposal under which we would be required to restructure the Company, hand over to Microsoft Yahoo!’s valuable search business and to Carl Icahn the rest of the Company, giving us less than 24 hours to respond. It is ludicrous to think that our Board could accept such a proposal.”

Microsoft sources consider the talks much less dramatic that that, noting that Yahoo had been talking to Icahn all week, and especially Thursday, about a new deal to take over Yahoo’s search business, and that an Icahn board was mentioned.

And, they add, Bostock was the one who insisted to Icahn that he needed to talk to Microsoft CEO Steve Ballmer directly, even though the software giant had publicly said it was unwilling to do so anymore.

But several times, the trio, along with a passel of bankers and lawyers, did just this, starting last Thursday night and, most significantly, Friday afternoon for 45 minutes.

It was in that call that the three sides discussed the proposal in detail, which included slides sent from Microsoft, although there was no term sheet.

Ballmer, a source close to both Yahoo and Microsoft said, did express being tired of the endless loop Yahoo and Microsoft were caught in and said that he did not want prolonged negotiations to go on.

This was not an ultimatum, according to Microsoft sources, but more of an expression of weariness at a new round of tedious back-and-forth.

“This has just gotten impossible,” said one Microsoft source. “We just wanted to have talks that went somewhere.”

When pressed, some sources close to Yahoo do confirm that they talked extensively to Icahn and then Microsoft. The 24-hour number came about because Ballmer asked Bostock to get back to him the next day, but that it was not exactly what one would call a threat.

“We felt he was saying Microsoft was not willing to put more on the table and, if we agreed to it, it would have been on their terms,” one source explained.

You get the picture–this is how wars start.

And that’s where Yahoo is now headed with Icahn–and by extension, Microsoft–as their proxy fight shifts into high gear heading into Yahoo’s annual meeting on Aug. 1.

What would it take to get Yahoo and Microsoft to come to some sort of rational agreement to strike a partnership of some sort before that?

Well, Sigmund Freud might be a start–or even Dr. Phil at this point–in this warped relationship.

Now, it seems, we’ll be moving directly to Judge Judy.

More on that later today.

Hopefully, we will find out this morning the 411 on the Microsoft’s (MSFT) latest parry in its attempted takeover of Yahoo (YHOO).

But, before that, let me get this straight: Microsoft CEO Steve Ballmer now wants to go hostile on Yahoo with a proxy fight.

Because earlier this week, he sent smoke signals that he was considering raising the price of his takeover bid for Yahoo.

And just before that, Ballmer was going to walk away from the deal. Except, before that when he was ready to lower his offer.

But don’t forget the friendly path Ballmer was also promising. Even though his initial unsolicited offer started off back in February as, well, hostile.

Calling Dr. Phil!

Perhaps the wacky TV shrink can merge together Ballmer’s multiple personalities into one, much the same way he tried to mend Britney Spears.

OK, that did not go so well. But it couldn’t be worse than the confusion of intentions from Microsoft here, including the latest one reported by The Wall Street Journal’s crack team today that Ballmer had now selected hostile as his choice.

Here is Ballmer’s money quote, the likes of which a Zen master could not interpret: “With the right circumstances it’ll happen. Without the right circumstances it won’t happen.”

What? That just about covers every option possible and then some. This may even require an Oprah intervention.

Here’s BoomTown advice, which we reiterate from yesterday’s post: Offer $33 to $35 a share as a best and final, take-it-or-leave-it offer. Set a 48-hour deadline for Yahoo to enter negotiations. After that, initiate a proxy fight.

This says a lot of things. It says Microsoft is generous (and will even bid against itself). It says it has had it waiting for Yahoo to dangle Google (GOOG) or AOL (TWX) as alternatives. It says definitively that Microsoft wants Yahoo and will fight for it.

Most importantly, it says enough is enough.

And while this may seem like it says a lot of things, it doesn’t seem–as so much of what Ballmer has been saying so far says–like the rant of someone in need of help.

Because, as it stands for Ballmer, that’s what it sounds like right now.