Ah, so the "over" bit is where McCoy in the briefing room argues against everybody's eagerness to attack? TrekCore and many other sources indeed have it wrong, but Chakoteya's transcripts (
http://www.chakoteya.net/StarTrek/9.htm) feature the correct phrasing. My fault for not checking!
(Then again, when McCoy explodes about starting a fight for "...memories of a war over a century ago!", he might not be exact or knowledgeable with his figures. He is, after all, a doctor, not a re-enactor.)
No, the ``over a century ago'' comes right from Spock's exposition at the top of the episode, in which he explains the war for all the people in the audience who weren't there earlier. The closed captioning transcript omits the ``over'' for some reason (possibly to make the text better fit the screen -- closed captioning started out as trying to simply summarize text, and it's grown more complete as broadcasters learn that yes, people can keep up with all that much text), but Spock says the war was over a century ago. Given the context of the Science Officer briefing the crew, I think we have to assume he's exactly correct about it being not a day short of 101 years.
http://tos.trekcore.com/audiocaps/1x14/1x14-tos-03.mp3 contains the relevant audio segment.
What gives you the impression of Kirk not knowing much about the old war?
A valid point: it's just an impression. But I find it quite odd that Kirk would react with apparent surprise to the fact that the Romulan ships had birds on them (that's what his reaction looks like to me, rather than general surprise at Stiles for stating the factoid).
Well, it does all come down to impressions. I don't think he's reacting to anything besides Stiles's confidence that the symbols of The Enemy wouldn't have changed in a century of unknown developments and changes on their part. (Besides, Kirk did seem to acknowledge the bird-of-prey was a common symbol of the Romulans a century ago: he relies on Stiles's knowing it to conclude that Stiles has greater historical knowledge than the run-of-the-mill crewman.)
I mean, that's about the only thing known about the enemy, so it should count for all the more. Could we forget that the Nazis were the guys who sported swastikas, or that the Japs were the ones with the rising sun symbol?
On the other hand, a person woke up from a century's nap in 1942 would have no way of recognizing the Germans or the Japanese by these symbols, since the Germans hadn't used them in 1842 and the Japanese, well, who heard of the Japanese in 1842?
When that's how Kirk launches into this thing, and then relegates the knowledge stuff to Spock, never saying much himself, one may argue he's playing the usual strong and silent type, speaking little but saying a lot. But one may also argue he just plain lacks the knowledge to say anything about the subject. It's not as if he ever offers any true nuggets of information there, apart from his orders not to cross the Zone.
Compare this to other instances where Spock (or perhaps McCoy or Chekov) gives an introductory speech about the background of the week's adventure for the audience's benefit, and Kirk then fills in the details, seemingly showing off his expertise. Say, "Friday's Child" or "Trouble with Tribbles" or "A Private Little War". Kirk is really being exceptionally silent in "Balance of Terror".
On the other hand, ``The Trouble with Tribbles'' does follow roughly this pattern: Spock provides the broad-picture exposition and then Kirk prods both into explaining the precise details of the current situation. In ``Friday's Child'' Kirk lets McCoy provide the background material on the Capellans and then mentions they're out to sign a treaty to get Macguffinite shipments in. In ``A Private Little War'' Kirk asks one little question about flintlock development and sets off an anthropological debate on the bridge crew that Picard would envy.
In ``Balance of Terror'', Spock provides the background of who the Romulans are and why there might be some irredentist feelings involved, and then Kirk steps in to ratchet up the tension, informing the audience that the ship just might be destroyed and the crew killed this episode, and it's not a hoax! Not an imaginary story! For real!
Then why was it called the "Earth-Romulan War" and not the "Coalition-Romulan War"? Mighty egotistical for the humans to take all the credit.
You expect war names to make sense? Consider the Franco-Prussian War, where Prussia was aided by both the North German Confederation and miscellaneous South German states (and where Prussia merged into Germany as a result of it). Or the War of Jenkins' Ear. For that matter, the French and Indian Wars aren't so cleverly named either. (Oh, now, wouldn't that be an interesting twist? I mean, if Earth and the Romulans were allies against some other force, and part of the terms of the war's resolution was the dividing line between the two powers? I suppose that's hard to reconcile with Stiles's fury at the Romulans, though it could be managed.)
Anyway, assuming that Earth and Romulus were the leading powers behind the war, then Earth-Romulan War is a reasonably sensible if unimaginative name for it.