There is an m:n relationship between meetings and formats, but this is not represented in the DB. Instead of that, meeting_main.formats is a comma separated list.
This not only makes the search code unnecessarily complicated, but results in a big performance hit for larger DBs (and, I assume, for aggregator).
Using Laravel's wonderful migration mechanism and ORM, this should be relatively straightforward to clean up. And, it's a pure refactoring, so we have the tests.