This is just a guess -- I don't remember reading it in any book -- but I think the exhalation might be to give the other cat a good sampling of their "internal smell" (for lack of a better description.) I've noticed that when something's wrong with one of my cats, the others sniff at that cat with worried expressions. I think they can smell another cat's general state of health, or unhealth. So the exhalation might be saying to the other cat: "I'm a strong and healthy cat."
My cats don't do the exhalation w/nose bump, so another guess would be that an open mouth is being misinterpreted as an exhalation when it's really an inhalation
: the cat is flehmening -- drawing air into a little cavity in the roof of the mouth called the Jacobsen's organ, which has sense receptors similar to smell receptors. (Yes, a cat really does have an extra sense we don't have.) So the cat is "sampling" you.
To add to what gunterkat said, face-to-face sniff & bump should be followed by a full-body side-swipe and then a sniff at the other's rear end. You can try that with your cats and watch their expression.
A cat doesn't like to have a big human face stuck right in their face (it could be interpreted as a challenge or as a threat if accompanied with a direct look) so you do the nose bump at the cat's leave. If you force it on them, they'll turn away, or bop you in the face. A bop is cat for "stop it" or "don't do that."