We can't answer why some people leave it out of some conversations. It's definitely present in anything that tries to be a comprehensive discussion of the Cold War — the "losing of China" was an important Cold War event, as was the Korean and Vietnam Wars, both of which had strong involvements by China.

China's place in the Cold War was complicated. They were initially very close to the Soviet camp, but in the 1960s they broke away from them (the "Sino-Soviet split") and became officially Non-Aligned. This puts them into a different category from the superpowers by definition, and lumps them with other non-aligned nations like India, Yugoslavia, Ghana, Egypt, and so on. Many popular histories of the Cold War ignore all other blocs beyond the US and Soviet Union, which is a major error and much ink has been spilled by Cold War historians to try and reconsider the ways in which non-superpowers exerted considerable influence on the direction of the Cold War (and upon the actions of superpowers). A classic text in this vein is John Lewis Gaddis's The Cold War: A New History.

Anyway, there is much that can be said about China and the Cold War, but they did not position themselves as a "third superpower," and telling a version of the Cold War that was just about the US, USSR, and China would be committing its own omissions (again, of places like India, or even European states that were ostensibly aligned with the US but certainly had their own agendas as well).

In terms of China's nuclear arsenal, they started a nuclear weapons program in the mid 1950s after the Eisenhower administration threatened to nuke them over the Taiwan Strait Crisis. They were able to develop warheads fairly quickly, but it took some time for them to develop weapons that had intercontinental range — their initial forces were only able to strike targets relatively close to them, and so either worked as a deterrent against the USSR (who were not their friends by this point) or as a form of extended deterrence against the USA (e.g., threaten US allies — like Japan — as a way to threaten the US by proxy). China only deployed true ICBMs in the 1980s.

The approach they used is what is sometimes called the Minimum Means of Deterrence, which emphasizes having only enough weapons to make an enemy think that it is not worth starting a nuclear war with you. For them that was, apparently, around 200-300 deployed weapons. That is a number that is low-enough that it would never make the US or Soviet Union think that China was in a position to try and execute a first-strike attack (they would never be able to destroy the thousands of weapons of the US or USSR before they were retaliated against), but it is enough to still create unacceptable harm in both countries (losing even 20-30 cities would be ruinous). As for why China took this approach and not the more maximalist one, I don't know if we really know the "true" answer. It appears to have been partially strategic (such a stance is avoids certain kinds of brinksmanship, and is inherently defensive), partially economic (nuclear weapons systems are huge drains on the economy, and during the late Cold War China was busy building a booming economy), partially ideological (they may have seen themselves as being more enlightened than the Superpowers), partially propaganda (they may have wanted to look, for both domestic and foreign propaganda purposes, like they were not participating in the arms race, and they consistently argued that theirs was only a defensive capability).

In any event, they appear to be changing their approach now, possibly because advances in US and Russian capabilities mean that the smaller Chinese nuclear arsenal is itself vulnerable to a first-strike attack (the US could potentially imagine destroying all or most of their forces in the future, and shooting down what did get off the ground), and so they are apparently expanding their arsenal considerably (to judge by the missile fields under construction).

Anyway, it is very difficult to know whether there are hidden, internal reasons for why various policies were taken in China around their nuclear program, because a) they don't say all that much about why they do what they do, and b) a lot of what they do say about it is hard to distinguish from propaganda. But if you are interested in what we know about the Chinese approach to the Minimal Means of Deterrence, Jeffrey Lewis' The Minimal Means of Reprisal is a very useful monograph, as his monograph Paper Tigers: China's Nuclear Posture. For a detailed coverage of the Chinese nuclear program in its early days, the classic text is Lewis and Xue, China Builds the Bomb.