Alice falsely believes that killing her husband Bob would be morally right (say, because Bob committed adultery). She shoots Bob, trying to hit him in the head, but misses completely and Bob escapes. Alice has committed attempted murder. But while she attempted to kill Bob, she did not attempt to murder Bob. For it was her intention to rightfully kill Bob rather than to murder him.
Therefore, attempted murder is not an attempt to murder. Speaking very carefully, we should say that Alice committed an attempted murder but did not attempt a murder. What she attempted was a killing, a killing that would have been a murder had she succeeded.
While the point is particularly clear in the case where the attempted murderer believes the killing to be right, the point also goes through in cases where the attempted murderer knows the killing would be a murder. Suppose Chuck wants to inherit an estate from his uncle Dave. Chuck knows full well that killing Dave would be a murder, and he attempts to kill Dave to gain the estate. Unless Chuck is especially malevolent, Chuck's intention is that Dave should die rather than that Dave should wrongfully die. After all, whether Dave's death is wrongful or not does not affect Chuck's inheritance (as long as Chuck doesn't get caught, that is). Thus Chuck did not intend that Dave be murdered, but only that he be killed.
It seems likely, thus, that in typical cases of attempted murder there was no attempt at murder, but only an attempt at a killing, a killing that the malefactor did or did not know to be a murder.
The point goes through for other misdeeds. An attempt at theft is typically an attempt to take something, but not typically an attempt to thieve. An attempt to lie is typically an attempt to convince of p, but not typically an attempt to convince of a (subjective or objective) falsehood (the crooked car dealer attempts to convince you that the car runs well, and that's all--she isn't trying to convince you of a falsehood as such).
Roughly, it seems that we call an action "an attempt at M", where "M" is a morally loaded description, provided that the agent is attempting to N, where "N" is a morally unloaded description, and where the N would be an M were the agent to succeed. But that's only a rough characterization. Here's a weird case. Erin has just picked up an alien weapon and is attempting to kill Frank, an innocent person, with that weapon. Unbeknownst to Erin, the weapon is a smart raygun that only fires at people whom it is just to kill (e.g., it checks whether the killing would be a part of a just war). Then Erin has committed attempted murder, even though had her attempt to kill succeeded, she would have been engaging in a just killing rather than a murder. I don't know how to characterize attempted murder to get out of counterexamples like this. An interesting ethics project for a graduate student!