OK, reporting back 
To clarify my thinking, I found it useful to distinguish between:
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Free content on Vudu.
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Content on Vudu that has been purchased by the user but is not in MoviesAnywhere.
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Content on Vudu that has been purchased by the user and IS in MoviesAnywhere.
On ATV 4K:
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Verified that I am logged into Vudu app and can access and play my purchased but non-MoviesAnywhere content while in that app, and also a purchased movie that is in MoviesAnywhere.
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Using Siri Remote: found the non-MoviesAnywhere movie that I had just tested in the Vudu app. List of options for where to play it includes various services (eg Hulu) but not Vudu.
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Using Siri Remote: found the MoviesAnywhere movie that I had just tested in the Vudu app. List of options for where to play it includes various services (eg Apple TV itself) but not Vudu.
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Using Siri Remote: found a movie I know is in Vudu Free. List of options for where to play it includes various services (including Plex) but not Vudu Free.
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Install JustWatch. List of services to connect shows Vudu Free but not Vudu, so I did not try playing anything.
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Install Reelgood. List of services to connect to shows Vudu (but maybe only Vudu Free?). When I select that service, message appears saying something like: "No direct link available". When I actually add the movie I know is in Vudu Free to my "to-watch" and press play on Reelgood on ATV, it says something like: "No app found for this".
And, a possible (partial) explanation:
https://discussions.apple.com/thread/251330346
That explanation is from 2 years ago, and suggests that the only linkage between ATV and Vudu is via MoviesAnywhere. So content that is not in a user's MoviesAnywhere will not show up or be playable in Vudu.
That still implies that it should be possible to play MoviesAnywhere content in Vudu directly. And I'm fairly sure that I at some point in the past used to be able to do a Siri Remote search and open links in Vudu (pretty sure it was for all kinds on content, not just MoviesAnywhere).
But that clearly does not happen now. Maybe something changed with the Fandango acquisition? Not sure when that happened.
Sigh... guess I will have to stick with PlayOn Cloud recording from Vudu.
EDIT: deleted my question about PlayOn recording quality from below, and posted it in a separate thread.