How are boosted decision stumps different from a decision tree?
The decision boundary in (4) from your example is already different from a decision tree because a decision tree would not have the orange piece in the top right corner.

After step (1), a decision tree would only operate on the bottom orange part since the top blue part is already perfectly separated. The top blue part would be left unchanged. 

The boosted stumps, however,  operate (as you mentioned) on the full dataset again, which can lead to different results.