If moral equivalence doesn't apply to your view, why use the word "murder," then, to describe the killing of a bird?
Why not? I felt like it. Use a different word if you want. I might deign to use a term like "murder" to describe deliberate killings that are not done to satisfy a vital need, like food, shelter, water, defense... that sort of thing. In other words, it is a deliberate killing that probably did not need to happen and was done "just because" - maybe out of anger, maybe out of spite, maybe out of callous disregard... any killings for stupid and questionable reasons.
The killings planned of Barred Owl may not quite fall under the murder header by such standard. A killing of other persons that serves the vital needs of other persons is... well, as I said from the beginning I get where these kinds of decisions come from and their scientific basis. This is no matter of spite or anger, selfishness or flippancy. It is often painful for those who work in conservation - who tend to be far less speciesist and anthropocentric than other humans - to make these kinds of calls. The question now is how to give back and make it worth it. And, perhaps, when to know to let go.