To unclog a bathtub drain the right way, you'll need to take the drain plate off. That's the little round plate that has the lever on it for opening and closing the pop-up drain stopper. It takes a little effort, but it's not too hard to remove and behind that plate, you have access to a straight drain pipe you can send the drain snake down. Check out the video above from This Old House for instructions.
The word "plumber" dates from the Roman Empire. The Latin for lead is plumbum. Roman roofs used lead in conduits and drain pipes and some were also covered with lead, lead was also used for piping and for making baths. In medieval times anyone who worked with lead was referred to as a plumber as can be seen from an extract of workmen fixing a roof in Westminster Palace and were referred to as plumbers "To Gilbert de Westminster, plumber, working about the roof of the pantry of the little hall, covering it with lead, and about various defects in the roof of the little hall". Thus a person with expertise in working with lead was first known as a Plumbarius which was later shortened to plumber.