In the OT, murder seems to be defined within the context of one’s community. In other words, killing a kinsman (i.e. a fellow Israelite) is murder, but killing one’s enemy is not. Thus, Exod. 20:13 “Don’t murder” refers to killing fellow Israelites. Notably, Moses killed an Egyptian (Exod. 2:12) but he is not put to death by fellow Israelites for it because the Egyptian is an outsider. When an Israelite killed a fellow Israelite, blood vengeance was enacted (to fulfill “an eye for an eye”). In this case, an appointed family member was to seek out the murderer and kill him—the penalty for murder was death (Gen. 9:5–6; Num. 35:6–33—which is actually dealing with accidental murder and avoiding blood vengeance).
In the OT, killing others in an act of war is not considered murder; in fact, failing to kill certain foreigners was considered a sin! Examples: Deuteronomy 7:1–5; 20:16–18; 1 Sam. 15:1–35. These texts should make us pause: genocide commanded by the Hebrew God?
As for comparing homosexuality with murder, Lev. 20:13 does state that if a man lies with a man as one would lie with a woman, both participants should be put to death. Thus, the death penalty is applied to both situations, but keep in mind that the death penalty was applied to many things, including blaspheming God, blaspheming your parents, committing adultery, etc. So, I don’t think it’s necessary to single one thing out over another when it comes to the way they applied death penalty back then.
All that said, the OT system of justice was very different from our system of justice. I think it’s a grave mistake when people try to use the Bible as the basis for a modern justice system. I also think it’s hypocritical for people to point out verses like Lev. 20:13 but ignore other verses in the same chapter, for example, “If a man sleeps with a woman while she is menstruating, both he and the woman will be cut off from their people” (Lev. 20:18, I paraphrased it). When’s the last time you’ve seen some moron holding a sign saying “God hates men who sleep with their wives during their periods”?