I don't understand everything he means by that, but how is China in a different "core" position than pre-war Germany was or then post-war Soviet Union was?
These are different times, in Barnett's view. Under globalization, the economic and commercial interests trump martial endeavors when in competition with other Core members. To some extent, nuclear weapons preclude war between Core members (but a nuclear weapon in the hands of a member of the Non-Integrating Gap is a different matter). I think that Barnett's 4-2-1 argument is an important observation. China is simply going to be too old to fight a war.
I don't know how he fits Taiwan into the picture. It is obviously a member of the Core.
Germany between the two World Wars and post-war Soviet Union were pre-globalization, although there may be some parallels between now and the 1930s, according to Barnett. Different rule sets applied in those times. He argues that the rule sets were evolving and that the security rule sets became obsolete in the 1930s. Yeah, it's wonkish, think-tank, Pentagonese to throw around ill-defined terms such as rule sets. If you've got an interest in trying to figure out the present situation, Barnett's book is interesting. I don't know if it has all the right answers, but it is an interesting take on our security concerns.