I might be showing my ignorance here… but I believe smores is correct. Institutional separate accounts are managed to their specific mandates. Similar accounts are grouped into a composite for performance reporting / marketing purposes, but my understanding is that a composite isn’t managed as an entity in and of itself like a fund.
What STL is describing sounds more like a retail SMA (wrap account) model portfolio.
That said, to answer the original question both above responses are correct. Indexes are calculated by index vendors, composites are based off of actual portfolios managed by investment managers.