In the case in question, the former. https://en.wikipedia.org/wiki/Anwar_al-Awlaki

Is there? One is actively killing citizens the other is still a reckless disregard for human life in general.

Is there a difference? Legally? Both would be official acts, right?

Why are actions being taken against military targets outside of a declaration of war?

Can you give an example of a time when a "legitimate military target" needed to be killed so immediately that a unilateral presidential decision that also killed nearby citizens was warranted? My presumption is that there are very few circumstances where the threat of a military target is so immediate that killing nearby innocent civilians is justified over waiting to try to find a better time to attack the target. If anything, immediate threats that require action are generally threats to the nearby innocent civilians themselves, which would make such decisive action not particularly useful; I don't think anyone would suggest something like blowing up a bank where people are being held hostage, which seems like the closest hypothetical I can think of.