I'm 46. One of my earliest memories is of a Mugato, and with my parents' help I can successfully pinpoint to first-run TOS. I have been listening to the MLK story my entire life.
I know that it's changed. I would be unable to give empirical evidence, but I know that if you asked me at different times in my life about the story, I would have different answers -- because the story was always changing slightly.
Going 'way back ... maybe to TMOST, which was published during the show's run, my infant and early childhood memories are unclear ... but ever since I have been aware that Nichols once entertained leaving the show, there has been a slightly different twist on it. Initially, it was all just the money.
My most vague memory is that there's some kind of race reason given ... I wonder if I'm remembering a pre-TMP-era interview somewhere. If I recall, it was in that "tween period" from around 1972-1977. Everybody, Roddenberry, the studio, the actors, knew there was money to be had from Star Trek. The actors would get memos from Roddenberry who'd say, "This is great, we have a start date but no script. But what the hell, most people have a script and no start date!"
So there was an interview, maybe in the context of one of the big East Coast ST cons of the 1970s. They got into talking about residuals. Everyone agreed, to the reporter's surprise, that they hadn't been paid since the third viewing. That was all a standard contract ever bothered to deal with, since it was utterly unheard-of for a show to do so phenomenally well in syndication.
Before anyone suggests I'm remembering the Tomorrow Show With Tom Snyder interview, I'm not. That had a similar conversation, but it was between Doohan, Kelley, and Snyder. The one I'm remembering was Nichols, Koenig ... and one of the popular babes of the week. They were at some con and had shot interviews with the local news. The popular babe of the week might have been Barbara Luna, but I'm not sure.
Anyway, they were being interviewed, talked about residuals, how they weren't getting any, not even Bill or Leonard were getting any ... the studio was making money but none of the actors were. It was one of those moments where you can see that someone feels like they're kind of getting screwed.
At some point, Nichelle mentioned off-hand wanting to quit after the first year because she was getting paid so little. She didn't mention MLK at all. She said that she didn't really have much going on otherwise and finally decided that at least where she was, she was showing a female black officer in a racially-mixed cast. Maybe that counted for something, she thought. Then I think she realized she might be coming off bitter and that never plays well to fans, so she started talking about how great the cons she'd been attending were and how wonderful the fans are, etc.
Thinking about it, I recall that story being re-told in the context of interviews surrounding TAS. It was the same story told a bit more positively, and with the spin of being so happy to be doing Star Trek again and doing voices other than Uhura's. The "moral of the story" became, "I once thought about quitting, but I was nuts. A female black officer on TV is a great role model! And look what I'm doing now!"
So the slightly bitter interview might have been 1972-73. The TAS retelling would have been about 1973.
The first time MLK entered the picture was years later. I want to say that it was in context of some friends who'd gone to one of the Star Trekons in Kansas City at which she was a guest. They brought back the local papers' coverage of it. I think it was when the cast was being groomed for Phase II. The story became, "I once thought about quitting, but I thought, 'What would MLK do'? Here I am, a black female officer, an equal among whites and men, how could I quit? And now, of course, we're getting called in for fittings and they're writing scripts and we're going to do it again. And I'm going to have a lot more to do than I did in the original series ..."
I want to say that the first iteration I ever heard that actually involved MLK personally must have been during the press junkets surrounding STTMP. The first iteration involved a relatively chance meeting ... the two of them happening to be in the same hotel, and it turned out he used to watch the show. They wound up in the same room (MLK asked to see her in some early iterations, I think). She said she was flattered he liked the show, but to be honest she was thinking of quitting. He advised against it because she was a black female officer and a role model.
After that, the specific details and changes get vague. It's never been that the story has been horribly embellished out of shape over the years. It's that minor individual details change over time -- but just frequently enough to jog your memory of the last time you heard the story.
It's also not entirely Nichols' doing, either. The moment MLK entered the narrative, the story grew a life of its own within fandom. It would travel more by word of mouth between fans. By the time I remember actually hearing Nichols invoke meeting MLK, fans had been saying to each other, "Really? Well I heard she actually met him!" for months.
I'm not quite sure whether this is part of aging that I enjoy or not: that I can just barely remember how stories have changed during my lifetime.