When we created visits (I think it was at our 2nd implementers conference around 2008), we defined “encounter” as a point in time interaction, “visit” as a collection of encounters that happen within a single contiguous event (e.g., outpatient clinic visit, hospitalization), and “episode of care” as a collection of encounters belonging to a specific condition or treatment (likely across visits).
Program was designed to enroll patients into a study or treatment program. Patients can be moved within exclusive states in a program so we could answer questions like “who are all the patients in the HIV treatment program with a status of ‘on treatment’?” Program (whether a treatment program or study protocol) also implies patients could be assigned a program identifier – she is patient 1234 in Dr. Doe’s Diabetes study.
In FHIR, encounter is used both for the OpenMRS’ encounter & visit. I would suspect “episode of care” could, for example, relate all encounters related to a single pregnancy, a course of TB treatment, etc. I would imagine an OpenMRS program could either be linked to an episode of care… or maybe it’s a specific use case for episode of care… but I could easily imagine uses for episode of care for which we wouldn’t want to require a program.