The concept of a rules-based international order is not bound to US (or any other) hegemony.
The term "rules-based international order" as used in that interview means that international order is regulated by a set of laws that are above any local legislation passed by members of interntional community. This idea is pretty much accepted by all politicians regardless of country, but the exact ruleset this order should be based upon is a subject of much controversy. It would be easy to define it as "the sum of treaties ratified by various countries" - problem is, there are no treaties that would be fully ratified by every single member of international community. Whatever treaties you choose to be a part of your theoretical ruleset, you will have to deal with all the countries that are not signatories to that particular treaty. The "Western rules-based order" mentioned seems to be based upon the treaties ratified by NATO countries, and thus the opposition includes a lot of countries that NATO has tensions with.
Hegemony is not a "rules-based international order" (unless the only rule of this order is "hegemon's word is law") - by definiton, hegemony is the fact of a country being dominant. Thus, American hegemony is a concept separate from a RBIO. On the other hand, a hegemony could use its political weight to make other comply to a ruleset the hegemony supports. But if a big enough subset of community has a ruleset they all agree on following, is able to enforce compliance to that ruleset on actors outside that community, and has enough political will to exert that ability, they don't really need a big bad bully to help with this.
Obviously, US government support is not for any abstract RBIO, but for a very specific ruleset. Just as obviously, opposition will argue that this ruleset is a tool to further USA political influence (and thus a pillar of "American hegemony"), and the rhetoric differentiation helps to position this set of rules as "for the good of humanity" and not "a tool of Pax Americana".
Looking at the question you linked, whether American hegemony and/or US-supported RBIO exist is a question which does not seem to have a clear, universally accepted answer. That means that anyone who disagrees to follow the ruleset undermines its existence, as international law is customary - basically "we do things this way because we always did things this way", and any noticeable (i.e. expressed by a politically influential country - like China) disagreement with a set of rules means this set won't become a part of this "always did" customary structure.
P.S. I couldn't find any statement in the interview that could be interpreted as your bullet point #2, so I did not address that.