No, that was Kirk's interpretation of the PD for that particular episode. Almost every time that the PD was ever brought up in ST was when Kirk was about to violate it based on his own interpretation of it at that particular time.
And it's your interpretation that he was violating it rather than upholding it. I think you're approaching it through a 24th-century lens, where they've taken it to the cowardly extreme of avoiding any involvement at all even when their inaction leads to massive destruction. That's not how TOS interpreted the PD. It wasn't about Starfleet avoiding responsibility at all costs. It was about respecting self-determination. When Kirk intervened, it was to cancel out other interventions and restrictions on free choice, whether from computers like Vaal or interlopers like the Klingons, John Gill, and Captain Tracey.
And again, I will say: too bad. The Federation isn't supposed to be Team America:World Police (or galaxy police, in this case).
Again, that's a TNG-era perspective you're taking. Maybe you could argue that Kirk's actions would be seen as inappropriate by Picard and his contemporaries. But in the era when he lived, and in the minds of TOS's writers, the Directive was not meant to be that rigid. Preventing alien interference in indigenous cultures meant preventing
anyone's interference. It meant that if something had already disrupted a culture, it was allowable to remove that disruption and restore the natural balance.
Of course, "A Private Little War" was not meant to represent an ideal scenario -- that's the whole point of the episode. It was meant to represent a Vietnam-like situation where there was no good answer, just the lesser of evils.
Yes, it sucks that other alien powers are subjugating lesser races, but really, what would have been the tactical advantage to arm some primitive people with flintlocks just because the Klingons intend to eventually subjugate them? Why were those people so important that Kirk was willing to "interpret" the PD in his own way? It looks far more like Kirk is doing this only because he seems to feel some personal attachment to Tyree, not because he's trying to uphold Federation law.
What was the tactical advantage to intervening in Korea or Vietnam? It's about not letting the enemy get a foothold -- a matter of principle as much as a move in the great game. Yes, the local population was reduced to pawns in that game, suffering as a result of the larger-scale struggle between superpowers, but the superpowers saw no choice. This wasn't like "The Apple" or "Return of the Archons" or whatever, because there was more at stake than the Prime Directive. The mindset of the Federation and the Klingons in the era was like that of the US and the USSR in the '60s, a belief that the very survival of their nations depended on not giving ground to the other. And in the context of that mentality, they were willing to compromise other principles like the PD. That's not about Kirk alone. That's about the entire astropolitical situation he was part of. Probably any other commander would've made a similar choice. How many times did America decide its laws and principles could be bent in the name of Cold War survival?
More generally, you have no basis for the opinion that Kirk's interpretation of the PD in other episodes was in any way unusual for the era. If anything, when you look at the other captains and influential figures we saw in the era -- people like Merik, Tracey, and Gill -- Kirk actually seems to be one of the more conscientious observers of the Prime Directive. Others violate it for personal gain or to impose their philosophies, but Kirk only intervenes when he deems it necessary to negate someone else's disruptive interference.
And the very fact that Kirk kept his command and even got promoted to admiral -- twice -- should be proof enough that Starfleet did not consider his actions to be in violation of Federation law. Which is why it seems likely that the 23rd-century Starfleet's interpretation of the Prime Directive was more flexible than the 24th-century version, and that Kirk's approach to it is actually pretty typical for the era in which he lived.
It was realistic enough in the Star Trek universe that the very same thing happened on another planet under the same circumstances, as my example of Mordan shows.
I could argue that it's not an identical situation at all, because the conflict was entirely internal and only one person provided weapons to both sides; but "Too Short a Season" is just such a terrible episode that I don't consider it an exemplar of anything "realistic."