You phrase "stops objects from passing through each other" is very similar to the one I use in the classroom. I use "resists" rather than "stops" because things can break or be forced through pores, or, or, or ...
It's worth emphasizing that it doesn't matter whether the objects are being slowing pushed together by some external force or simply move free into each other (collide). As long as they would have needed to occupy the same space, the normal force comes into play.
Now, you seem to be a bit confused on Newton's 3rd law and its relationship to normal forces.
Newton's law says that
When body A exerts at force (any force) on body B there is at the same time an equal and opposite force on body A from body B.
That's true for any1 force, including the nromal force.
When applied to normal forces Newton's law just tells you that both bodies experience the normal force at the same time, with the same magnitude and in opposite directions (which is good for preventing them from occupying the same space, they are pushed away from one the interface between them).
1 The form of the rule I have written here isn't actually true for all forces, but there are other version that are true for absolutely every force. Don't worry about the exception to this version yet; just keep them in mind for later on.