The West Bank and Gaza were under the control of Jordan and Egypt, respectively, since the 1948 Arab-Israeli War. But they were captured by Israel during the Six-Day War, and controlled by Israel ever since. It does not appear then that any Palestinian government was involved in the execution of the Six-Day War.

This is correct, but only somewhat. There were figurehead Palestinian "governments" that existed, but they operated out of Gaza, not the West Bank. The "All Palestine Government" for example, recognized by the Arab League (minus Jordan), was nominally in charge of Gaza and the West Bank, but operated entirely under Egyptian military authority and held no real power. Thus Palestinian governments were almost nonexistent.

That's not to say that different groups didn't vie for a spot as Palestinian representatives on the world or at least regional stage. The PLO, for example, was created by the Arab League as such a representative, though it was largely coopted by Egypt before the 1967 war. Other Palestinian groups carried out guerrilla attacks on Israeli civilians and forces before the 1967 war (more on that later), vying for attention/support, but were largely vassals or at least reliant on sponsor states (Fatah, for example, was heavily supported by Syria in its early days). Thus these groups certainly existed and vied for attention/power, but they were at least prior to 1967 still reliant on larger Arab state patrons for support, and were certainly not governments despite their proclamations to that effect. The PLO, after all, angered Jordan by declaring that it viewed Jordan itself, the West Bank, Israel, and Gaza as all part of Palestine and to be recovered by the Palestinians, effectively suggesting that Jordan had no right to exist either, and that the PLO should rule at minimum the West Bank.

Did Palestinians organize and fight along with Jordan and Egypt? At that point, was there a meaningful distinction between Palestinians and Egyptians and Jordanians, or did Palestinians still consider themselves occupied, such that Palestinians could disassociate from the Jordanian and Egyptian governments/militaries?

There was a distinction militarily, but as I mentioned, many of the militias at play were effectively vassals of larger Arab states. On the Jordanian front, the PLO had attempted as I mentioned to become more influential. Of course, this was met with derision...whenever Jordan was complying with Egyptian wishes. As recounted by Michael Oren in Six Days of War, Shuqayri (the PLO head who had pledged to lead an army to Jordan's capital and ignore King Hussein) claimed he was head of the Palestinians and demanded recognition, saying he wanted to travel to Jordan. Nasser, Egypt's leader, laughed and told him he was not going to Jordan anytime in the near future and told Hussein to throw Shuqayri into one of his towers and get rid of him if he gave him trouble. At the end of the day, Palestinian leaders were either nominally independent, guerrilla groups coopted by the Arab states but given some range of freedom, or entirely subservient to Arab states.

The PLO was more than happy, on the Jordanian front (Fatah was far less welcome) to threaten Israel. Indeed, a few days before the war, the PLO's Shuqayri was giving a fiery sermon when he included that the PLO was "prepared to take its place in advance positions on the Jordanian front so it can stand face to face with the Zionist gangs".

This was, of course, par for the course. Both the PLO and Fatah had, among other smaller Palestinian groups, launched raids on Israel prior to the war. Fatah's goal was explicitly, as recounted by Moshe Shemesh in "The Fida'iyyun Organization's Contribution to the Descent to the Six-Day War", published in Israel Studies to "ensnare" Israel in a war with the Arab states. Fatah at least hoped that launching these raids would lead to a war that the Arab states would win. Destabilizing the border was one route towards doing so. As a result, the Arab states had varying approaches to Fatah and the PLO, reining them in sometimes when it seemed war was becoming too likely (and cracking down), and sometimes letting them loose. The groups often despised that they were being held back to avoid war, but as I mentioned above about the PLO, states like Egypt could still effectively leash them. If anything, Palestinian groups were more eager to attack Israel than the Arab world, which was largely still uncertain it could win in early 1967 (despite public protestations to the contrary).

The PLO did, in fact, have some forces and participate in the war. However, they were only nominally under PLO control, as with everything else; instead, they fought as the Palestine Liberation Army (PLA) and were organized and integrated into the Egyptian and Jordanian fighting forces to some extent.

From what I understand, Palestinians were occupied by Jordan and Egypt and then caught in the middle of a war which resulted in them being occupied by Israel. But what is troublesome is that, if the Palestinians had no ability to exercise their right to self determination while occupied by Jordan and Egypt, then is it fair that Israel considers the Palestinians territories to be occupied as the result of a defensive war, which is a grey area in international law, if the Palestinians were not belligerents?

Israel doesn't regard the Palestinians as having had a claim to the territory to act as belligerents. There was not a prior Palestinian state, they argue, which means they cannot be occupying the Palestinians. Israel does not consider the territories occupied at all, it considers them "disputed", which is meant to reference Israel's position that the area was uniquely stateless and messy due to the end of the Mandate. It argues that any self-determination claim to the territory by the Palestinians must be balanced with Israeli claims under Mandatory dissolutions that happened during the period, referencing legal principles such as uti possidetis juris. That is the idea that once a Mandate is dissolved and independence achieved, the successor state receives the full territory and invasions are illegal. That argument would, if accurate, mean that Israel has a claim (though not the only claim, in theory) to the territory of the West Bank and Gaza as it was the only state that declared independence in the former Mandate, though it didn't declare itself a successor state, weakening that claim. Israel also argues that it has a right to defensible borders and that the belligerency of Palestinians in 1948 weakens their claim; Jordan's annexation and occupation of the West Bank was illegal and the result of its invasion of the Mandate/Israel, which means Israel doesn't have to recognize the Palestinians as belligerents.

Generally, however, Israel's arguments are not accepted outside of some prominent legal scholars and jurists, though most disagree. At any rate, I can't necessarily answer whether it's fair for Israel to consider Palestinians occupied via defensive war, because Israel doesn't consider itself an occupying power in the sense commonly thought of. However, Palestinians did indeed fight in the 1967 War. Whether those fighting on their behalf constituted a force to represent them is another matter; you then have to get into questions of what creates "representation" for the purposes of war (it certainly isn't democracy in most of the Middle East during this period, since wars were waged by dictator leaders too), what level of independence from other states is necessary if any for that "representation" (it clearly isn't full independence, as some of the Arab states were effectively under Nasser's thumb at some points or another, and Lebanon is a case study in unclear independence from foreign influence, some argue), etc. etc. Given the complications, and the fact that the question doesn't necessarily bear weight on the actual Israeli position, I hope I've helped at least a bit!