It’s mostly summed up in the last paragraph of the post that you linked. Basically:
If governance allocates coretime to a chain AND its upgrades/root origin are deferred only to system governance, then it is a system parachain;
Everything else is not a system parachain.
Within the category “not a system parachain”, people can call things whatever they want, and perhaps it does get subdivided into multiple classifications of chains (or more generally, coretime consumers). If people want to adopt “common good” as a term with specific meaning, then go for it . However, if you do want it to have specific meaning, someone should express that more precisely.