Bridgeit, I think this is one of those enjoyable ambling debates that everyone likes putting their six penneth into.
My personal take as someone born in Devon, raised in Bristol and since then has spent the last forty years living in North Somerset, I feel the terminology used by locals is somewhat different to those living outside the region in making named reference to the area.
Persons such as myself in the main have always referred to the area to the west of Gloucester as "the West Country" as that is where the Severn estuary begins splitting the west of England peninsula away from South Wales.
However, as I stated earlier, many holidaymakers travelling down the M5 and M4 motorways begin to believe they are in the "Southwest" as they pass over the M5 Avon Bridge South of Bristol and see the Severn estuary from there and also from the following "split level" section of the motorway. From that section on a clear day you can see Exmoor in the distance especially if they are sat in one of our famous M5 traffic jams.
Therefore I believe there are two terminologies used for the area to the west of a line from Gloucester to Weymouth, one being "The West Country" with the other being "The Southwest Region".
I prefer the West Country, but if you are one of the ever growing numbers of holidaymakers down here, you pays your money and you takes your choice.