They want their side to win?

A lull in artillery fire would generally support one side or the other. I don't think that generals' interests systematically differed from those of their sides in such a way as to make it in both sides' generals' interests for their to be artillery fire rather than no artillery fire.

I think it is more likely that each was mistakenly overconfident that they could win with their strategy and tactics. WWI is replete with examples of generals' overconfidence and I think that is a much better explanation of why each would prefer his orders followed than wanting to "maintain the killing"/"supposed to be fighting", which seems like suggesting they are innately evil.

There are many examples of officers thinking the war would be best conducted under their offensive strategy when all were wrong and a defensive strategy would have been better.

They want their orders to be followed?

This is a very good point.

I still don't see any sign of cooperation among generals across sides. I think it is more likely that each correctly thought the truces bad for morale, and the Central Powers were concerned their troops would cease to have the morale for offensive operations while the Entente was concerned that their troops would cease to have the morale necessary to fight at all. This would leave some wrong about who the truces favored.

When two executives are each trying to get their organizations to defect first in a standard prisoner's dilemma when the organizations both have impulses towards cooperation, the identical strategies among the executives do not constitute "cooperation" between them.

I agree each has motivations to see that their orders are followed regardless of what is good for the organizations, but this is not cooperation because both would have their organizations defect regardless of whether the other organization was inclined to cooperate or defect.

Cooperation between generals on two sides might look something like the Battle of Tannenberg, where each Russian general may have wanted the Germans to beat the other general before beating the Germans himself. That would be manifested as lack of conflict, such as the Russian First Army waiting for the German Eighth Army to beat the Russian Second Army before joining the battle.

Cooperation to induce fighting might be each HQ broadcasting vulnerable locations for the other artillerists to shoot at, or similar. I don't know of any such cases, but then again, I wouldn't necessarily know of them.