Well, to further confuse matters, there are areas of oceanic crust on the continents, such as in eastern Panama, which were uplifted from the ocean floor. So what are those, islands? Oceans? In fact, it doesn’t make any sense to classify landmasses as continents or islands based on the composition of the crust. “Continental crust” was named that because it makes up most of the continents; to work backward and classify islands as continents merely because they are made of the same kind of rock has no particular logic to it. It doesn’t mean that landmasses made of continental crust should be called continents. If you have “no patience with arbitrary classification schemes,” you’re not going to improve matters with the one you suggested.
The thing is, the idea of what a continent is has evolved over time. In the ancient world, they were simply broad geographical areas distributed around the Mediterranean, which happened to be separated by minor seas (Red and Black) or the Strait of Gibraltar. It didn’t matter whether or not they were surrounded by water, or connected in some distant region like Europe and Asia were.
It was only with the discovery of the Americas that “continent” began to take on its modern meaning. People became aware that these landmasses were similar in size to the known continents. At the same time it also became evident that Africa was mostly surrounded by water. “Continent” began to be understood as referring to a major land mass mostly surrounded by water (even though Europe and Asia were broadly connected). It should be noted that the classification of what is and is not a continent varies culturally; Latin American geographers (and I believe many Europeans) consider the Americas to be a single continent whereas we divide them into two.
Once Australia and Antarctica were discovered the term “continent” was extended to them as well, based on the new concept. (It helped that they were both completely isolated from the other continents.) Finally, this concept was applied retroactively, and many geographers began to lump two of the classical continents together as Eurasia.
In deciding the final number of continents, I think there has been a tendency to consider that they must be comparable in size to the original three continents. Hence there is some ambivalence about Australia, which is slightly smaller than Europe. Greenland and other large islands are clearly much smaller, and so have never been considered continents.
The result is that we have a semi-scientific overlay on what was originally a purely arbitrary classification. This prompts people to try to make the definition even more scientific. Personally I think this is rather pointless.
The progression in classification from classical to semi-scientific is echoed in the classification of the planets. The ancients knew of five planets, which were simply stars that moved. The Earth was not one of them. Once the nature of the classical planets became known, the original concept was changed completely so that it included the Earth. The definition has changed and evolved further as the asteroids and Kuiper belt objects have been discovered. The first asteroids discovered were considered planets, as was the first Kuiper belt object, Pluto. As more and more such objects were discovered, astronomers tinkered with the definitions in order not to have “too many” planets, resulting in the recent demotion of Pluto. But these are still arbitrary classifications, even if put in scientific terms.
