Well, she did try to back away before fainting, so it's probably the latter.
Picard and Data are touching the Phoenix with their hands, and Picard is reflecting on how it makes the ship seem "more real" while Data sees no difference (due to his emotion chip currently being inactive). Troi comes up to them, unseen:
Troi: Would you three like to be alone?
Deanna: Time line?! This is no! time! to argue! about! time! We don't! have! the time! (beat) What was I saying?
Riker: (smirking) You're drunk...
Deanna: No, I'm not!
Riker: Yes, you are.
Deanna: Look. He wouldn't even talk to me unless I had a drink with him. And then, it took three shots of something called "tequila" just to find out he was the one we're looking for. And I spent the last 20 minutes trying to keep his hands off me! So don't go criticising my counselling technique! (beat) It's a primitive culture... I'm just trying to blend in. Riker: You're blended, all right. Deanna: I already told him our cover story. He didn't believe me! Riker: We're running out of time. Now if we tell him the truth, do you think he'll be able to handle it? Deanna:If you're looking for my professional opinion as ship's counselor... he's nuts! Riker:I'll be sure to note that in my log. (Cochrane restarts the jukebox, blasting music, while Deanna collapses, making Riker facepalm.)
What makes it even funnier is that Riker was actually enjoying the music before Troi faceplants into the table.
Her Punctuated Pounding is a very directed poke to his chest, which really just makes the conversation. Riker's expression during it as well, going from "I don't believe this!" to barely-holding back laughter, not-quite-succeeding with his customary grin.
According to director (and cast member) Jonathan Frakes, Drunken Troi is worth the price of admission alone.
And when Riker's not dealing with Deanna's drunkenness, he's dodging the various objects that Cochrane keeps throwing in his direction.
And just before, this part;
Cochrane: This is the good stuff. To the Phoenix! May she rest in piece.
During a part of the time that Riker is asking if Cochrane can handle the truth and Troi says that he's nuts, Cochrane is standing right there.
Cochrane's initial reaction to the situation:
Cochrane: Hot damn, you're heroic! *laughs*
After Cochrane becomes convinced of the truth, Riker goes on to explain how the approaching aliens will detect his warp flight and "land right here."
Cochrane: Here? LaForge: Actually, over there.
A large portion of Zephram Cochrane's scenes, but especially:
Geordi:(engages in long spiel about Cochrane's magnificent standing in 24th century history, and the giant, inspirational statue of him that will stand right where he is standing now) Cochrane:(beat) ...I gotta take a leak. Geordi:Leak? I'm not detecting any leak. Cochrane: Don't you people in the 24th century ever pee?
Its also funny as part of Trek's running gag about the TOS Enterprise apparently not having bathrooms.
There is something hilarious in Cochrane's motivation speech. Everyone is making him out to be this hero and visionary, and he's just in it for the money. But especially hilarious is when he explains that he really shouldn't be doing this:
Cochrane: I don't even like to fly!
Cochrane's exclamation of "SWEET JESUS!" when the Enterprise casts its shadow over the Phoenix.
Also, Cochrane's hilarious "AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAHH" when the Phoenix finally breaks Warp Factor 1.
You can see Geordi and Riker grinning at each other behind him.
The Defiant Conn officer having a subtle worried glance when Worf states that "Perhaps today is a good day to die," followed by the order to ram the Borg Cube.
When the helmsman tells him the Enterprise has arrived, Worf almost looks disappointed that his Heroic Sacrifice has just been preempted by a Gunship Rescue.
After the Enterprise is ordered to the Neutral Zone, the scene between Picard and Riker is brimming with Sarcasm Mode.
Riker: We finished our first sensor sweep of the Neutral Zone. (hands Picard a PADD) Picard:(reading) Oh, fascinating. Twenty particles of space dust per cubic meter, 52 ultraviolet radiation spikes, and a class-2 comet. Well, this is certainly worthy of our attention.
When Data keeps asking questions of the Borg Queen, she asks — with perhaps a touch of exasperation — "Do you always talk this much?"
Data: (Beat.) Not always... but often.
Thinking about it, it's also a pretty awesome moment - Data's babble managed to get under the skin of a collective of techno-organic emotionless zombies. That takes SKILL.
Picard being snogged by a ditzy blonde in his Dixon Hill holodeck program, making you wonder what he really gets up to when he's not being the upright Starfleet captain.
Picard, Worf, and Crusher briefly note the Enterprise's short life after the activation of the self-destruct and wonder whether there will be another..... Then Picard points out there are plenty of letters left in the alphabet.
Data revealing to Picard exactly how long he considered the Borg Queen's proposal: 0.68 seconds. Even if it is "almost an eternity" to an android...
First contact itself. The Vulcan gives the "V" salute, Cochrane attempts to, then goes in with a handshake instead. Cochrane's verbal reply is both funny and heartwarming:
Vulcan: Live long and prosper.
Lily gets a hold of a phaser and warns Picard that if she doesn't start getting answers, she'll start pushing buttons. Later, Picard finally manages to talk her down and gets the phaser away from her.
Picard: Maximum setting. If you'd have fired this, you would have vaporized me.
Lily: (meekly) It's my first ray gun.
Picard, Lily in tow, finally reunites with the others, including Worf.
Worf: (seeing Lily's slightly-panicked stare) I am a Klingon.
Reginald Barclay's Adorkable fanboying over shaking Cochrane's hand...
Barclay: I mean, meeting the man who invented warp drive...
Lily tells Picard to get her off the ship, not realizing where she actually is. So Picard opens the porthole and shows her, pointing out the various features of Earth...
Picard: Montana will be up soon, but you may want to hold your breath. It's a long way down.
The interaction between Lily and Picard is generally very serious and tense, but it's easy to get a giggle out her half-snarky, half-bitter response to Picard's smashing the glass on his display of model ships, sending them toppling.