Yup, 2, 3, and 4 are absolutely the Trek Trilogy, but Kirk's arc through the movies starts in TMP and goes through to Generations. This was mostly accidental, I think, but it works well.
In TMP, Kirk is still fairly young. It's not long after the five year mission, and he's now an Admiral. But he hates it. V'Ger comes along, and rather than him wishing Captain Dekker bon voyage, he essentially steals the ship from him. Kirk is an absolute dick to Dekker in that movie, and McCoy directly calls him out on it. Kirk's life had moved on from being a Captain, but he couldn't let it go. He has the command back at the end, but at what cost?
2-4, as you mentioned is pretty directly a trilogy, with Kirk reborn and reinvigorated as a Captain of a new Enterprise. That new energy carries directly over to 5. The first you see of Kirk, he's free climbing El Capitan, McCoy is watching him, grumbling about how stupid it is. We get Kirk musing about death around the campfire. He's accepted death as inevitable, but it's not real to him. He lost Spock, but got him back, but at the cost of his son. He's accepted the "no-win scenario"…but not his own death.
In 6, Kirk finally deals with the loss of his son. He's able to make peace with the Klingons, and at the same time, peace with the inevitable. He finally has to move on from being Captain of the Enterprise.
Then, in Generations, he has the chance to take command once again, but unlike in TMP, he doesn't take it. He lets Harriman be the captain, while he goes to save the ship, and "sacrifices" himself. Finally, the no win scenario was that he had to die to save the ship. He gets sucked into the Nexus, and we see Kirk at peace. He's out of Starfleet, married, chopping wood and making eggs. It's only when this punk kid Picard comes along and guilts him into action that he decides to come out of retirement one more time to save the day.
It's not the best write up, but that's how I see Kirk's arc through the movies. Probably the most complete character arc in all of Trek.
