Some suggestions of changes in language to be more accurate to terms used by systems:
A "core" is an "original", in systems where there is someone considered the original.
A "host" is the main fronter, usually being the one with the most control over life and body. May be the core, may not be.
When someone is in control of the body, they are fronting.
Alters cannot be "manifested" (i.e. go in and out of existence) but they can front infrequently and only in situations where they are considered to be needed.