Well your definition seems very narrow. What if two aircraft are merging on a convergent course, and one turns away slightly to avoid the HO / ram and the other shoots. Is this a HO?
Secondly when engaged already with one or more cons, and another one comes into the fray, you have insufficient airspeed to pull G out of a guns solution and get shot in the face, is that a HO?
I think yes in both cases (and know many who think likewise) and in both cases only one aircraft has a guns solution.
Think of head-on attacks as jousting. Two knights (so to speak) sinking spurs and aiming their lances at each others chest.
In your first example you are describing someone avoiding or attempting to avoid a head-on attack. So we have one pilot attempting and one avoiding. At that point, there's no joust. If the attacker presses and manages to hit, it's a failed maneuver on the defenders part. The key word is defender. That pilot went on the defensive. A head-on attack (joust) involves no defensive technique applied.
(Let me amend this. Bombers can be attacked head-on. This is generally not a mutual thing.)
Your second example involves a situational snap-shot. The attacker that is entering a furball is not capable of keeping his guns looking down the sights of any one specific target. The other planes are maneuvering already, they are engaged. If he manages to hit a plane 'in the face' it is because they turned their plane and the bullet or cannon rounds struck at that moment.
Now, if a plane managed to disengage from the fray and head straight for the lone pilot heading into the fray and they both line up sights on each other to fire (whether both fire, then, or not) ... that is a head-on engagement.