Using plate tectonics to define continents does have some interesting effects:
India, the Arabian peninsula, The Caribbean, and the Philippines all become separate continents.
We have a number of sunken or submerged continents (plates with no major land masses above water, only some islands): the Pacific Plate, the Nazca plate west of South America, the Scotia plate at the bottom of the Atlantic Ocean, and the Cocos plate west of Central America.
Also, Greenland and the eastern part of Siberia would be part of North America.
As far as I know, though, only geologists use plate tectonics in this way.
Maybe we could do it this way: the plates that have landmasses could be called "continents", and the plates that don't have landmasses could all be called "Depends" because they would be "incontinents".