What is this type of car called?

Roadsters were and are basically vehicles without roll-up windows and were manufactured by many car companies. Convertibles do have roll-up windows and Cabriolets had roll-up windows with a structural mass.
California cruisers of the 50's and up are vehicles which many chopped or just removed a sedan top and went from there. Some added Carson tops, a Carson top is just a home made top which is removable yet adds no strength, no windows and is basically for show. Back in the mid 60's I had a E Jag with welded in doors, no top or windows and a cut and lowered windshield yet had a roll bar and structural appendage which surpassed the normal street car for safety. The car was similar to any race car of that era but street driven.