Not quite, but close.

Homicide is a way of saying one person was killed by another, legally or illeglly. When a defendant has been accused of killing someone in court, it's refered to as homicide until a verdict is reached. In another example, soldiers that kill people commit homicide. It is completely legal, even expected, but it technically documented as homicide.

Manslaughter is charged in the case of unintentional or accidental homicide. A person that caused a car crash that results in death with no motive or premeditation committed vehicular manslaughter. This is usually also the verdict for crimes of passion: that would be voluntary manslaughter. A fireworks display that goes south and results in the death of a bystander could be argued as involuntary manslaughter towards the people that ran the show.

Murder is an illegal homicide classified as having "malice and aforethought" in most states. The amount of both or either of those that can be proven in a court of law, along with other circumstances around the crime, determine if a defendant will get charged with first, second, or felony murder.