it's a shorthand to say "Mastodon" since no one really knows about the stuff under the hood (ActivityPub and the federated social web)
Yes everyone (especially those taking public money) should build AP into their existing web infra like their content management systems.
it's a complete failure of policy that all of these public status communications are made into centralized services with hostile ownership. for this, I mean your local police department and fire department, library and mayor and all of those groups.