The Biblical notion of faith is never a mere belief in God's existence.
Even "belief in God's existence" is more than a notion - a notion is a belief of postulation - a possibility.
A belief is something held up as essentially true - like I believe my hands are attached because I'm using them - whereas if I considered it a likely that the Broncos will win on the weekend, my belief is that they might - my belief is in a possibility, rather than a concurrence of my reality. As with faith, "belief in" is more than a notion of possibility - belief in a possibility is what a belief in a possibility is.
---------------------------------------------------------------------------------
Words to convey are used in such a fleeting fashion - the term "faith in God" would best be written "a notion about the existence of God" if it were the intended meaning, and likewise - "belief in God" would better be written "belief in the possibility of a God" if that indeed was the intended meaning.
These phrases describe something much more tangible than an accepted notion or a postulated philosphy.