I watched that scene recently. I think I've always interpreted it as Kirk being so hyped up at just avoiding the ship's destruction. that he pressed it twice just as a reflexive instinct signifying that he had so narrowly avoided the end. He knew that it wouldn't reengage the destruct sequence, in fact, by itself wouldn't have initiated or caused any action to take place. I've often done something similar in repeating a movement out of relief of having gotten out of a bit of a ticklish situation. Just sort of a release of a built up anxiety being avoided, that's all.
Also, perhaps it's been addressed at some point, but the so called Halkan prediction of galactic revolt has remained a burr in my side. It's never mentioned by the Halkans, as we see their interaction with the landing party, and yet without it, how does Kirk start his memorable dialogue with Mirror Spock at the end? I guess one can only surmise that the Halkans mentioned it prior to the scene as we saw it, otherwise how can you reconcile this gap in continuity for something being propounded, that even at the time of the episode itself, must have left viewers with the distinct impression that changes would be forthcoming for Mirror Kirk, let alone that they much later became the reality in the franchise and novels? It just seems that a rhetorical point of such import, even in the realm of the episode itself, just was poof! created out of nowhere at an especially propitious moment.
Another possibility just occurred to me that might very well make the above statement moot. It actually wasn't necessary for Kirk and his party to have heard this themselves from the Halkans. Aside from anything else,, that would imply that the Federation was going to be overthrown, the rationale for which would seem a little harder to swallow. What makes imminent sense, and again may long have been the accepted truth around here about the incident, is that Kirk gleaned the mention of the threat from Mirror Kirk's personal logs, as they might very well have made an initial visit to the Halkans before the one in which the transferral took place. That would certainly makes sense, although again for the audience, without having actually heard the warning themselves, they would have a bit of work to do figure out the scenario.
Has this latter interpretation been widely agreed upon to explain the origin of the statement?