We know that in TOS era Star Trek command officers wore gold and operations officers wore red. In the DS9/TOS cross-over episode Trials and Tribble-ations O'Brien explains why Sisko (pretending to be a Lieutenant) is dressed in gold:

BASHIR: Wait a minute, aren't you two wearing the wrong colour?

O'BRIEN: Don't you know anything about this period in time?

BASHIR: I'm a doctor, not an historian.

SISKO: In the old days, operations officers wore red, command officers wore gold.

However, all of the footage of Captain Kirk in the episode shows him wearing green, not gold. Here are two images from the episode, the first shows Captain Kirk with the tribbles:

The second shows Sisko talking to Kirk, and it is clear that Kirk's uniform is a different color from Sisko's gold.

So, my question is Why is Kirk's uniform green in this episode and not gold?

Apologies if this is well known to TOS fans, but my Star Trek knowledge is TNG and DS9 focused. I could not find an answer after some quick searching.

The Command Tunics in TOS Were Actually Green

The costume designer for TOS was Bill Theiss:

William Ware Theiss - the designer of all the Starfleet uniforms and alien costumes seen in the 79 episodes of The Original Series. His elegant, daring and revealing wardrobe creations for the female guest stars on the series is one of the most memorable and iconic aspects of TOS.

The blog Startrekpropauthority has the full text of an interview with Theiss in 1988. In that article the color of the command uniforms is discussed:

Another quirk involving the original series’ tunics were the colors - in particular, “command.” Trekkies everywhere will swear Spock wore blue, Scotty wore red and Kirk wore gold. Wrong. The three Starfleet colors were blue, red and green. Lime green, to be exact. “It was one of those film stock things;” Theiss states, “it photographed one way - burnt orange or a gold. But in reality was another; the command shirts were definitely green.” As further proof, look at the wrap-around tunics as well as the dress uniform tunics of Kirk’s – all green. They came off as their true colors because they were constructed of different materials than the standard duty command shirts.

There is an additional article on the command tunics. Apparently there was a change from Seasons 1 and 2 to Season 3:

For the third season of Star Trek: The Original Series; Starfleet tunics and dresses were fabricated from a nylon diamond weave double knit fabric that would not shrink during cleaning - which had been a persistent problem with the velour material used for the first two seasons. The actual Command tunic color used in the 3rd season was Lime Green; which photographed as Gold under the lighting conditions of the Desilu soundstage.

Here are some images of the regular command tunic, from two different photoshoots (I believe these are the Season 3 costumes, images from startrekpropauthority):

This is not a duty uniform.

This is a captain's casual / off-duty tunic. Notice in your bottom-most screenshot, that the tunic is split down the center with one side overlapping the other, as in a cardigan.

Here is a better screenshot from a different episode:

Compare this to Picard's captain's jacket (worn on occasion in TNG Seasons 5-7):

It's a byproduct of (poor) restoration.

Fabric type does not change how film renders color. RCA was not trying to promote color receivers with oversaturated colors in the programming; they had enough trouble building cameras and receivers that performed consistently.

Film stock of the nineteen-sixties was easily able to accurately render lime green and lime green and gold as gold provided that the cameraman and the lighting director knew what they were doing.

Color TV was as new to production crews as to audiences. Directors tried to get images that looked "natural" on the equipment at their disposal, in some cases all production was done in film, and viewed as film. The color movie production standard was to err on the side of oversaturation, because film stock colors dilute during projection and fade with age. (Acetate film stock also tends to yellow as it ages.)

What was not capable of accurate color rendition were the color cameras used in the film to video conversion and the television receivers. Decisions on what colors should be used for costumes were modified after it became obvious that the original colors did not render well through video.

Variations in custom colors and spots skin tone are due to bad restoration technique. Leonard Nimoy's makeup had a faint green tinge to it and was supposed to render that way in contrast to William Shatner's slightly ruddy makeup. If restorations had been done from film, and directed by someone familiar with the original series, it is likely that the colors would look significantly different everywhere,not just the costumes, the actors, the scenery, everything.

If you look at the backgrounds of the restorations now being broadcast, you will see that all foliage is tinted toward cyan, less yellow, more blue than natural, while all flesh tones have been muted and normalized. This is the result of subjective decisions, not technology.

At the time Star Trek was produced, no one dreamed that the programs popularity would endure, that the series would be a cash cow for decades. It endured not because of the technology used to produce it, but because if the future it tried to portray. The special effects and stilted acting style of the original series are laughable today,but were acceptable to the less sophisticated viewers of that time. What endures are the relationships between the characters and the values and attitudes they portrayed. They showed our better natures becoming norms, not ideals.

People making sweeping generalization here are making a big mistake because you have to qualify which season uniforms.

First and second season velour command duty uniforms - also worn by Chekov and Sulu - were more of a antique gold color while some of Kirk's alternate uniforms were green. There are surviving Star Trek velour costumes from the original series in private collections that prove that fact and you can find pics of them around the web. It was gold; not green.

From the third season they used a double-knit polyester material and these the color was a khaki green which appeared tan/gold on film especially after many film transfers with degradation in the film color. The third season polyester uniforms actually do appear more green in recently restored DVD releases.

Kirk is special. It distinguishes him more. Green is cooler. That is all...