I've seen both theoretical and observational definitions of stars. For example, an AGB star is a star where two sets of nuclear reactions (helium to carbon and hydrogen to helium) are taking place in shells around the core, and a super AGB star is one that is also burning carbon in the core.
Is there a similar theoretical distinction between red giants and red supergiants? I know RGB stars are burning hydrogen in a shell while the core is inert, so is a red supergiant a star which is also burning helium in the core (or something similar)?