Help me settle a bet!
Well, colehankins, you are helping to prove my earlier point that this concept leads to some justifiable confusion.
First, the definition of "lee shore" was established to be any shore that is on the leeward side of a body of water (as Jeff stated). In your example, the west sides of both islands would be lee shores (with the wind out of the west, as you implied).
With your boat in between the islands, yes, you would also be "in the lee" of the island to your west. And if you ran east out of the lee of the first island, you would be running onto the lee shore of the second island.
Your last point is where I always see the trouble: people standing on the side of the island where the wind is blowing from will not think of that as a lee shore. To them it is the windward shore, and I don''t blame them for thinking that.
I attribute the confusion to the peculiarities of the jargon. In this case, the exact phrase you use ("in the lee" or "lee shore") mean opposite things.