I should apologize for the uncharacteristic text wall.
I fall on the side of Kirk not being a by-the-book kind of Captain. Not that he was a total maverick, but he was willing to do what he thought was right regardless of the rules. When the situation called for it I think he violated orders more frequently than other Captains. Based on this Memory Alpha article about the Prime Directive
someone on Reddit
put together a chart comparing TOS, TNG, DS9 and VOY breaches of the Prime Directive. TOS led with twelve and it only ran for three seasons compared to the other three that ran for seven. I didn't verify the work in the Reddit post but it seems about right to me.
There's some debate on what the Prime Directive actually entails, and if it meant the same thing right down the line. Originally it was about not interfering with the natural development of cultures, that were presumed to not be spacefaring themselves. Over the years people have folded into that the concept of not interfering with the internal matters of an advanced culture which was part of the Articles of Federation, and not the Prime Directive. The internal politics of member worlds were their own issue.
In TOS Kirk was generally dealing with a situation in which someone else had already messed with a less advanced culture, leaving him to pick up the pieces.
"A Private Little War" - The Klingons were supplying the villagers with firearms, before the Enterprise arrived.
"Patterns of Force" - Cultural observer John Gill started the Nazi movement in order to help in his own survival, by stabilizing the planet's society. The Enterprise crew had to undo the damage.
"The Omega Glory" - Captain Tracey had already mowed down thousands of Hill People with a phaser and messed with the local culture, by effectively making himself a warlord.
"A Taste of Armageddon" - Eminiar 7 was already a spacefaring culture, aware of others, who chose to be insular and kill each other. They went to war with the Enterprise and Kirk fought back.
"Bread and Circuses" - Merchant Marine Captian Merik told the legate(?), Claudius, about The Federation. Another mess for Kirk and company to clean up.
I'll certainly concede that the following were direct violations of the Prime Directive:
"The Feeders of Vaal" - For better or worse, the Feeders chose to worship a machine at some point.
"Return of the Archons" - Once again, for better or worse, a society chose to let a machine rule over them. It got logicked to death.