The dictionary defines "war" as a state of open, armed conflict between nations, states or parties.

Based on this definition, it seems to me that:

1. Since the Star of the West was, arguably, on a hostile mission (at least as far as the South Carolinians were concerned), she was unarmed and, therefore incapable of "armed conflict".

2. The Tyler had no military or government connection whatsoever and no armament, so that incident doesn't qualify.

3. The innkeeper, though probably armed, was not an official representative of any nation or state.

So it looks like Mr. Gorgas wins -- the bombardment of Sumpter was the first incident in which one state/nation sanctioned group attacked another.