Deadweight Loss and Excise Taxes
The problem is that your supply-demand graph is wrong. You assume in your graph that the demand and supply are linear functions but in this case they are step functions. 

For example, in your graph there would be 2.5 demand at price $\$0.55$ but the problem clearly states that the units are sold at discrete quantities of 1. So the drawing is incorrectly modeling the situation as there should be a steps in increases in demand not a simple linear function. The same holds for supply.

If you draw your supply and demand correctly you will see that in your case deadweight loss (DWL) is not a triangle but just two squares with areas 0.10 respectively which combine into a rectangle with area 0.2 and hence total DWL will be also $\$0.2$.

This is how your problem would look visualized (pardon the bad drawing):



Where the shaded area is the DWL. The two methods will always match but you have to use the same demand and supply functions in both problems if you assume different supply and demand of course results will change (in your original drawing also consumer surplus after tax would be wrong by the way as in your original drawing it would be area of two trapezoids both having areas of 0.15, and the pre tax consumer surplus in your picture would also not be 1.2).