My current job title is "Data Engineer" I'm not sure how that differs from a "Data Scientist".
My Job basically aligns with Type A above. I mostly work on optimizing our industrial process through a combination of modelling and simulation work. Other than that I do quality/defect investigations when we have issues with defective batches etc and I do yield optimization work. I also oversee various plant trials as required.
I use Business Intelligence software (like SAS and COGNOS) but I use other tools as well (including my own C/C++ code). I lean heavily on my own theoretical knowledge - in particular metallurgy and minerals processing. (I am a Materials Engineer by qualification). I think most BI people would lack the background theory to do my role.
My Job title was more or less arbitrarily chosen by my manager (Other people in my team have 'Automation Engineer', 'Process Engineer' etc. as their role titles). I consider myself an Engineer above anything else.