People get on television all the time and say Arab when it is actually Arabian. I refuse to see why this incident is very significant, espicially since the British are called Brits, Australians Ausies and New Zealanders Kiwis.
I've never heard an Arab refer to themselves as Arabian before. In fact, most Arabs don't live in the Arabian region, especially if you define that as the Arabian penisula.
It's true that Arab culture dominates the middle east. The exceptions to this are Iran (mostly Persians although there is a significant Arab prescence, especially in the SW), Turkey (Turkish), Isreal (although the Palestinians are mostly Arab and outnumber the Isrealis, a few of whom are also Arab), some of the Lebanese (but most are Arab) and the Kurds (although they are sometimes considered Arab, this can be a touchy subject however). Syria, Jordan, most of Lebanon and Iraq are all considered Arab but wouldn't usually be called "Arabian"
However, many people's of Africa consider themselves Arabs. This isn't a technicality, Egypt alone has more population then the Arabian penisula. The African countries on the Med are all Arab (Egypt, Libya, Algeria, Morocco) and Arab culture extends somewhat inland, for instance some of the Somalia population would consider themselves Arab. So, you'd be better off describing it as an African culture then an Arabian one although that too would be way off base. Geographic terms don't summerize the term succinctly at all as it's a very broad and diverse cultural movement.
So anyway, what you think is a reasonable statement about someone else's culture can in fact be completely wrong. Saying "Arabian" would probably just confuse, not offend. But it shows pretty clearly why it's nice to be respectful rather then assuming what a label does and doesn't mean to someone else.