In English, "elder" is purely comparative, and of people. It does not imply that the person is, subjectively, "old". So, "I have 2 sons; the elder is 3 now." But "I have 2 uncles; the elder is 90 now." OR "I have 2 uncles; the older is 90 now".br Does Russian make this distinction?
Старший only means that the person is older than some other person (it may also mean that a person is in charge of other people of the same speciality in some company).
Старый, on the other hand, means "old", which, depending on the context, means either "not young" or "not new".
Could this also mean "My older brother is already going home?"
I recall one discussion section where дома was described as an adverb meaning "at home" or "homeward", so that я хочу дома meant "I want to go home" = "I want [to be] homeward (bound)"
It seems unlikely to me to mean that, but I thought I'd ask.