No, funeral is fine as long as there's a body. The next part is the graveside service or burial (or interment, entombment, cremation, inurnment, etc.), but that doesn't have to happen on the same day.
If you need a nonspecific formal word for what eventually happens to the body, you can use 'committal,' though it's not very common.
If you need to add a sentence in the death or funeral announcement explaining that people will not be going to the cemetery, you can say something like 'Private graveside services to follow,' 'Burial to be held at a later date,' 'Burial to take place in family cemetery in Norway,' etc.
Without the body, it's a memorial service, which can be held any length of time before or after the actual funeral, or in lieu of a funeral.
A wake is the vigil held before the funeral, and/or the party held after it. The word isn't used in all religious traditions; Irish wakes are well known, but you wouldn't really expect a low-church Protestant to have a wake, only coffee and sandwiches after the service.
Never heard of a remembrance vigil, it sounds like a paraphrase of something else.
Mourning service is also not idiomatic in my experience, more a cobbled-together description.
A ceremony is any formal ritual in or of the church: a wedding ceremony, baptism ceremony, funeral ceremony ...
Obsequies (plural) is an old-fashioned word for the funeral and everything related to it. I would be inclined to mark it archaic.