My reading of scripture suggests a different combination.
"Soul" is difficult to define from scripture, but it appears to me to reference the combination of mind and body excluding consideration of the spirit--which would be the operational state of an unbeliever, a person operating with the functions of just mind and body.
I see us in scripture as a combination of a rational, intellectual mind controlling a non-intellectual body in the material realm and a non-intellectual spirit that exists in the spiritual realm.
If our minds were fully in connection and control of both, we would be like Jesus, able to operate in both realms with full effectiveness.
But because of the Fall, we were born with "comatose" spirits, as unto dead, that are awakened when we are indwelt by the Holy Spirit. Until then, our minds are connected only to our bodies and we have grown up responding only to our bodies. But with our spirits now awakened, we can become more responsive to the "groanings"--the needs and impulses--of our spirits and put the needs and impulses of the body under the control of our minds.
Something you've said is important to this discussion: The Triniity consists of three Persons of one Substance. We as humans, regardless how we dispute the meaning of "body," "mind," and "spirit," are definitely not comprised by three "persons."