If you are the sea, is the beach, hai tan, if you are in the river or the river this can only call beach, sha tan. Use contain mean, sand, sha tan contains the beach.

Well in my opinion, "beach" refers to "海滩" or merely means "滩" more precisely. The explanation of "beach" is: The shore of a body of water, especially when SANDY or PEBBLY sloping down to the water of a sea or lake. So "沙滩" is just the first form of beach: SANDY and thus, u'r right, "沙滩" more like a sandy beach while "海滩" more like the ocean or sea but not equals a sandy cover. And one more tip for u, 海滩 and 沙滩 r sometimes the same in the general conception of chinese that most of 海滩 r sandy. They'r changeable in some circumstances.

海滩：一定是在海边的沙滩。 沙滩：只要是beach就一定是沙滩（有沙子的滩涂地）。因为在河边的沙滩有时候会说是河滩。所以你可以认为beach的意思等同沙滩。

I prefer to 沙滩。 沙滩 describes 沙（sand）for emphasis. 海滩 describes 海（sea） for emphasis. But these two words have few differences in my opinion.

In Chinese, sha tan or hai tan are pretty much the same, when we use them, they all refer to beach, for this sentence: "a beach on an island" I prefer to use sha tan, which makes the sentence is dao3 shang4 you3 pian4 sha1 tan1. However you can also use hai tan for this sentence, people will know what you mean, but for me I just feel more comfortable by using sha tan here, I guess it just depends on the way that people speak :)