{The} four Americans{,} who had returned from the island{,} fell ill

Solution 1:

None.

None of your sentences above give any clue to the fact that there are more than four Americans. Although four sentences are different, their differences' purposes are as below:

By using "the," you assume that your audience already know what four Americans you're talking about. Without the "the," it could be any four Americans.

Four Americans went to a restaurant. (There are four Americans, and we don't know who they are)

The four Americans who just went to Canada came back yesterday. (We know exactly which four Americans, among the 331 million, we're talking about)

The comma around your modifier phrase("who had returned from the island") simply changes this modifier's importance. Without the commas it is considered necessary information, and with the commas this information is considered unnecessary to the main idea and may be omitted. In either case, the meaning shouldn't change — it's the emphasis that does.

Jasmine, my cousin, is 14 years old.

Jasmine my cousin is 14 years old.

As you can see, none of these differences contribute to explaining that there are actually more than four Americans. To elucidate this, you can simply write:

Four of the Americans who had returned from the island fell ill.

This way, it is very clear that these four are actually four of a larger group of Americans, and since we only say that these four fell ill, the rest of them being healthy is therefore implied.