So, we have a nice apples and apples or apples and oranges comparison since we have the same leader, the same foreign policy challenge and so on, and only one big difference--going from minority government to majority. The Auerswald and Saideman book in progress (July as an end-date?) does focus on such things as the nature of the government, so on one hand, we expect that this might make a difference. However, we also find in the book that minority governments are not as paralyzed as people would aver, as it really depends on who bears the onus of cooperating. That is, does the government need to get some part of the opposition to cooperate? In which case, the government is weak and restrictions are likely. Yet if the opposing parties must cooperate among themselves, then the government may actually have a free hand.
Thus, it should make a difference if the government is majority or minority. BUT our book has a second step in it--when individuals are empowered (as in Presidential systems and single party parliamentary governments), the personality of that individual matters. Perhaps Harper himself is not a big fan of the combat mission, especially since his control freak-iness is constrained when it comes to the fighting.* So, to have more control, he might be inclined to stay out of combat.
* There are big risks if a leader does micro-manage a war effort--as he/she then becomes directly responsible for anything going wrong, and Afghanistan is a place where lots of things can and do go wrong--IED's, prison breaks, corruption, etc.So, I am less sanguine about the chances of Canadians being sent to southern Afghanistan than the allies. But as a social scientist, I am keen to see how the experiment plays out.