2 seat, 2 door convertible (as opposed to a GT car with has a fixed roof) designed primarily for agility and motorsports, not necessarily outright speed. Companies that cut their teeth building sports cars for sports car racing classes are the ones to look to for definitions.
Notice that Ferrari made sports cars that were open top cars and when they grew a roof they became GTs and said so in their names. Ferrari also reserved 2+2s as GTs as well. Porsche also know that the roof makes a car a GT and thus has it in the name of the performance variations of the 911 (GT2, GT3, etc).
Look at MG and Triumph. The convertible MGB was the sports car. Give it a fixed roof and it became a GT (and said so in the name). Spitfires were convertibles. Give it a roof and it became a GT as well. Look at Alfa and the GTV.
Calling a car a GT instead of a sports car is not an insult that should be avoided, as the GTs were often faster than their sports car stablemates, due to better aerodynamics and better rigidity.
Finally, let's stop with the 2 seat convertible sports car is automatically a roadster nonsense as well. Roadster is a distinct bodystyle. A roadster is a 2 door open top car without rollup side windows. Give it roll up side windows and it's a cabriolet/convertible (or if you are British, a drophead coupe). Often there were roadsters and cabriolets in the same model line. For the Sports car crowd, the most noticeble is the Jaguar XK120/140/150 line where you could get a roadster, a drophead coupe, and a fixed roof coupe in the same model line.
Roadster:
Cabriolet:
All of this came from early days of coachbuilding where there were many bodystyles to choose from for each model car. There were 4 door cars without rollup side windows and they were called phaetons.
Her's another example of roadster vs cabriolet in the same year and model:
cabriolet (rollup side windows):
Roadster (no side windows):
And Phaeton:
Cobras are roadsters. So are early Viper RT/10s. Caterham 7s are roadsters (and their Locost bretheren). So true roadsters do exist, even now (Ariel Atom is a roadster). Miatas are convertibles, but not, even with Japanese engrish marketing, roadsters. They are, however, the quintessential modern sports car.