There's a difference between intent and motive. Intent can often be inferred from forensic evidence, like if you ran someone over with your car over and over again, rather than just hitting them by accident. Whereas motive is nearly impossible to prove, though unfortunately our courts don't really care.

Killing someone intentionally vs unintentionally is absolutely relevant. But why you killed someone should only matter insofar as having a motive makes you more likely to be guilty as a matter of fact, not insofar as whether a given motive is admirable or contemptible.