Dandruff is a colloquialism it's not a medical term. Many people use it when they have not received a diagnosis, so no it is not synonymous with SD. Simple dry scalp = xerosis, that is not the most common cause of a flaky scalp by any means SD is. You can do more harm that good by treating for SD if you actually have irritant or allergic contact dermatitis.
Are you seeing the dermatologists with freshly washed hair or three days hair? Have they taken a scrape and stuck it under the microscope? Have you combed your scalp in front of the derm? Any family doctor should be able to diagnose the most common scalp disorders, they certainly do here in the UK. NC cannot/ should not amateur diagnose something three derms have failed to. Be aware that it is fairly common for two dermatological conditions to co-exist, sometimes on the same part of the body.
The one thing that can help all of the possible conditions you might be suffering with is diet and lifestyle modification. Overhaul your diet so it is anti inflammatory and nutrient dense, particularly look at your intake of oily fish and sugar/ white refined carbs. Balanced nutrition strengthens the skin barrier, calms any inappropriate immune response, reduces systemic inflammation, alters the amount and composition of your sebum.
If there is any chance it is SD be very careful what oils you are using on your hair - oleic acid is a total no no, you are best off without the saturated fatty acids like stearic and palmitic too. For several of the other possible conditions sulphate free can be beneficial because sulphates damage the skin barrier at concentrations as low as 1% (shampoo is ~20%). Ditto any products with an alkaline pH, skin's happy place is pH ~5.5, hair ~4.5.