if it is on-demand, then it is AVOD. A company that started as FAST might have pivoted or decided to release the content they used to program their FAST channel as AVOD as well.

I've personally been involved in doing this very thing, but just look at the apps for like Max where they have their linear channel offerings within the same UI as their VOD. While Max isn't ad supported, it's a similar concept.

Licensing is distinct for FAST and (A)VOD. That is, a service may have acquired the rights to show a specific title on FAST but not AVOD (or vice-versa).