This question arose from discussion on Should we rename or remove the "beta" label?. Sites fall into three main categories:
Young betas that are still very much in beta. Consensus seems to be that these should be labelled "beta".
Long-running betas that probably aren't going to graduate. That's fine, but people have pointed out that "beta" isn't really a good label for a four-year-old established site.
Graduated sites. These are the sites that have higher privilege levels, custom designs, elected moderators, and other differences.
"Graduated" implies that only this last category has really "made it", but SE has now determined that the second category is a reasonable end state that no site should be embarrassed about. So talking about those as "long-running betas" or "niche sites" or whatever, in contrast to those other guys who actually made it and graduated, feels a little off. We do need to talk about the differences between the second and third categories sometimes, so we need some label.
Is there a better way to talk about "graduated" sites? Maybe, since these are the ones that get custom designs, they're "promoted"? Not getting a promotion feels different from not graduating. Is there a better term?