People do realize the reason Shatner's "Khan!" scream is remembered is because most people think it's funny. Seriously, how much mileage have comedians and sitcoms gotten out of it in the last thirty years?
Most people don't consider it some dramatic moment that rivals the best other movies have to offer. It's remembered because Shatner is a ham and the scream was right up his alley.
Yup, he just screams for no reason, script, plot or otherwise.
I'm of two minds on the subject of the original Kirk scream. On the one hand, years of reviewing the scene has left me a little cold, it's easy to see why people think that it's funny or pointless but, on the other, when I think back how the scene made me feel at first viewings
I find it was and is still well conceived.
Kirk is reacting to Khan figuring out his plan, not taking Kirk's bait; "...if you're going to kill us you're going to have to COME DOWN HERE!" Khan then responds with "I've done more than kill you, I've hurt you, and I plan to go on - hurting you. I'll leave you as you left me, etc..." Then Kirk screams and we witness Khan having his emotional orgasm over the echos of Kirks scream. Khan won, at least for the moment.
Regarding nuSpocks scream, I liked it just fine the first time. The narrative flowed smoothly into it so I didn't feel jarred at all. I wasn't so happy with it the second time, though. Which imo demonstrates the importance of the First Viewing Experience. It worked the first time as intended so I will forever credit it a successful scene just as I do with TWoK.