Tom:I like the name!
I was going to suggest that the five-position situation should be considered both before and after the move. However, on reflection, it is very often the case that the fact of a capture is much more important than whether an empty triangle is made at the same time. So I suggest that it should be assumed that no capture is made for a Spightonian. This is pretty much what you have done, except that it would eliminate the final Spightonian, leaving 20.
(Sebastian:) You are right that when captures come into play, the distinction about what was there before becomes a bit moot. We could write this in the introducting paragraph. However, I would rather keep the name "Spightonian" for all 21 geometrically possible patterns - just as the periodic table ends with elements that hardly ever are realized.
Tom: Is there a reason that you haven't included the 'throw in'?
(Sebastian:) "throw in" seems to refer more to a tactical purpose of a move rather than a specific move shape itself. Could you draw the Spightonian for what you have in mind?
(Sebastian:) How do we include moves at the edge of the board in this list?