The word I used, "Scandinavian", can be used both to mean "part of the Scandinavian peninsula" and "part of or relating to the culture originating on the Scandinavian peninsula". Calling Iceland Scandinavian easily satisfies the second definition.
You're right, in that Scandinavia is a peninsula and Iceland being a far off island is not part of it.
English is a weird and inexact language where there are a lot of different things words can mean. Seizing upon a single obviously-incorrect definition is not the whole story.
I'm sorry this bothers you; however I fear that you face the same battle as would a German trying to get Americans to call their country "Deutschland" instead of "Germany".
The term Scandinavia includes Denmark, which is not on the peninsula.
The use to include any country other than Denmark, Sweden, and Norway, is something that will tend to really annoy Scandinavians, as we're generally consistent in that use and it's foreigners trying to tell is what names to use for ourselves.
It may be a losing battle, but it's not one we'll give up on.