The weights are not "source".
In the phrase "the model is open-source in every sense of the word", that, IMO, must include the weights.
The former type of data I wouldn't expect to ever be open-sourced. The latter type might or might not be, depending on the intent of open-sourcing the related system.
If I created a human language translation system that used a SQL database to store the dictionary (domain) data and claimed the system was open-source without shipping the domain data, I think people would rightly say that the system was not fully opened.