I suspect that the Demora intro was written for Chekov, but the "nurses" line (and possibly Scotty's about Kirk's chair) was clearly written for McCoy.
In the October 1 draft with all seven TOS characters, there's no Demora (or equivalent) at all. I was going to say Harriman is the only named Enterprise-B crewmember, but while they're referred to only by position in the stage directions, two others get named in dialog (the helmsman is "Mister Brooks" and the science officer is "Commander Kline"), so the whole concept of Demora was added after the cast was pared down (though the only other draft I have is the March 16 final version, so there could've been an intermediary version where it was Kirk, Spock and McCoy instead of Kirk, Scotty, and Chekov).
To summarize the differences, there's an initial scene with all seven of the TOS crew packed into a turbolift on their way to the bridge. Spock makes a dig at McCoy, ("I haven't felt this close to you in years"), and Kirk bitterly says they're being displayed like trophies, and Bones muses about what they'd be for ("'Outstanding Starship Captain,' 'Most Noted Doctor,' 'Best... Ears.'")
Harriman comes off a lot worse in this version, arrogant, bordering on asshole ("I see you found your way to the bridge. I'm sure you're all very tired from your tour. If you'll have a seat, we'll get underway"). The semi-named Science Officer gets the line about meeting living legends and reading about TOS in grade school. Kirk and crew immediately sit down, there's no milling around the bridge or chatting with Demora or anyone asking if Kirk's having trouble with retirement. Kirk still gets pressured into saying "Take us out." McCoy teases that it "was
very good, Jim," but Scotty's "tear to my eye" line is absent.
Since the tour was implicitly before the movie started, we come back after the launch to the TOS crew just sitting on the bridge, incredibly bored. They get the distress call. In this version, Kirk tries immediately to take over the ship, hopping into the command well, telling the helm to lay in a course and ordering engineering to fire up the warp engine before Harriman shuts him down and reminds him that he's the captain, now. The "Is something wrong with your chair?" line isn't present at all, the whole bit of business with Kirk repeatedly standing up then stopping himself replaced him unthinkingly swaggering into the captain's chair. Again, it's possible it was first added in an intermediary draft I don't have access to, but I can't prove if it was written for someone else or whom the way I can with most of the other lines.
They reach the Nexus. There are three trapped ships, instead of just two. McCoy asks "What the hell is that?," Spock answers the he can't tell, and asks to take over the science station, which he does. Kirk asks about the tractor beam, gets told "Tuesday." The first ship explodes, McCoy asks how many people were aboard and Spock answers. Kirk quietly tells Harriman to pull in and beam the survivors off the other ship while they still can, as in the final movie, though Harriman didn't invite his advice in this version. The second ship is destroyed before they get close enough to do anything.
The navigator reports their engines are being interfered with by the ribbon and they're going to have to drop out of warp. Scotty is outraged at the idea of dropping speed in the middle of a rescue, and goes down to engineering to make sure the warp drive stays online.
They get to transporter range, Harriman asks Dr. McCoy to go to sickbay to help out, McCoy asks how big the medical staff is, he finds out, "Tuesday," so he grabs the first person he sees ("Chekov. You've just become a nurse. Let's go.") and heads below. Kirk is getting annoyed that everybody else is taking their old job back except him.
Not-Tuvok reports that he's having trouble getting a transporter lock, so Uhura moves over to help, ("Honey, a transporter's just like a man... if you want him to work for you, you gotta boost your gain and modulate his signal.") Spock has the lines about the passengers on the stuck ship phasing out of the spacetime continuum, with a couple more Spock-isms (he opens with "Fascinating," and after Kirk asks where they're going, he starts to reply "Unknown. Perhaps a scan of the —" before being cut off by the transport exploding).
Kirk asks Uhura if they got them, and she reports they got 47 out of 150. The Enterprise gets snagged, and the semi-named helmsman is killed by an exploding console, so Sulu takes over piloting the ship. Spock has Demora's line about being caught in a [TECH] from the edge of the ribbon.
The scene in sickbay is pretty much the same, except it's got Chekov and Bones. McCoy mentions they've all in neural shock, and then after Soran gets knocked out while he was ranting, Chekov asks what he was talking about and McCoy responds, "I don't know, dammit. I'm a doctor, not a mind reader." McCoy is the one who finds Guinan and comforts her.
Harriman's stern bravado has shattered, and he remarks that he didn't expect to die on his first day as captain. Kirk tells him the most important thing about being a captain is how to cheat death. Kirk and Spock have the entire exchange he has with Scotty in the final film about how there's no known way to get out of this, but Spock has a theory, photon torpedos, Tuesday, keep her together, yadda, yadda. All the dialog is unchanged, except Scotty got Spock's lines and Demora got Hikaru's.
There is one important change in that scene, Harriman is confused by Spock's deflector dish plan, and Kirk just brushes him off with a "We'll explain later" and goes straight to deflector control. There's no bit where Harriman volunteers to do the modification and gives command to Kirk, but Kirk decides to give up the chair.
There's a little additional color as Kirk works on the deflector.
SPOCK: Captain, a certain alacrity in your efforts would be helpful.
KIRK: What happened to that famous Vulcan patience?
SPOCK: It does have its limits.
Spock orders the deflector activated instead of Harriman, Sulu reports that they're breaking free instead of Scotty. After they get out, Sulu reports that they're clear, Spock asks Uhura for a damage report instead of Harriman asking Demora. Uhura has all of Demora's lines about the damage, and Spock has Scotty's trying to reach Kirk on the comm, before telling Uhura to have McCoy meet him in deflector control. Same scene as in the movie, with Spock and Harriman looking out into space, then McCoy running up and asking if anyone was in there. Spock's response is a flat, "Yes."
There's another scene, that was also in the shooting script, where they talk about having searched the ship and nearby space without finding Kirk. It would've been another awkward replacement, though. In the the first draft, it ends with Scotty saying he never thought it would end like this, and Spock observes everything must end, but in the final, the lines are swapped to Chekov and Scotty. Even if Scotty may have expressed the sentiment, he shouldn't have done it in Spock's idiom.
Overall, the final version is an improvement across the board. Even with a few stretches of chracterization with Scotty having a theory and Chekov having medical training, the "Keep things together until I get back," "I always do" exchange makes more sense with Scotty than with Spock, Demora is a better continuation of the TOS legacy than just having the old crew upstage the new, unnamed one (and beats out poor Hikaru having only a couple of purely functional lines that are only two or three words long), Harriman comes off better being overwhelmed because he's green and then, to an extent, rising to the occasion, rather than being a buffoon who totally falls apart and literally gets brushed off by Kirk when it's time to save the ship, and Kirk comes off
way better being able to restrain himself rather than immediately trying to force his way into the center of the action, and in how he gets offered the chance to be captain again but demonstrates his maturity by passing the torch of his own free will, which caps off a character arc that goes back to TWOK, TMP... even "The Naked Time."
