Public Access Role edits record and saves, but record does not update

That's certainly odd, because if the outside non-signed in EOTI user does not have edit access then the URL will fail to open at all. I presume when the user saves the record, it seems to save OK with no errors. 

Is there any chance you have a pipeline running that could be unchecking that check box?  Or a Form Rule firing "when the record is saved"?

One diagnostic thing you could try would be to put in a notification to Fire whenever a record changes just to see what is happening.