I would say that the ERD diagrams that we did as an exercise are actually physical models. Most of them targeted a very specific database platform - PostgreS - and were just a result of a reverse engineering the existing schema and then making it look good. Logical models do not have a notion of a Primary/Foreign key - they show named entities, key platform independent attributes and relationships between entities.
I would also say that it is wrong to say that the same logical model (OMOP CDM) will have the same ERD and physical implementation for all databases. So yes - you are correct that each database would have to have an ERD implementation of the logical model. They will be similar but not the same
