Using the edges of the board in Go
Firstly, note that it doesn't matter who owns what territory until the game is over, and none of these are realistic ending scenarios. However, let's assume the game did end with each of your first two boards. 

On the first board, the entire board would be considered White's territory.

On the second, among two good players, Black and White would agree that Black's stone is dead. White would then automatically capture Black's stone before scoring.

What happens if Black refuses to agree his stone is toast? You do indeed have to play it out and end up with board 3. However, White is not penalized for playing those extra stones:


If you are playing with area scoring, stones on the board are a point each just like territory controlled, so board 3 scores just as well as board 1 for White.
If you are playing with territory scoring, all play after the game has ended is considered purely hypothetical - it exists solely to resolve the dispute over which stones are still alive. White proves her claim that Black's stone is dead by playing until board 3, but then the game reverts back to board 2 (and White auto-captures Black's dead stone) before scoring.