The axe might be better in secondary mode from horseback...
you lose a percentage of damage and pierce gets a bonus on head shots.
All attacks get a bonus on head shots. Damage type doesn't matter at all (except for armour penetration).
Hitting someone with 30c on the head gets the same bonus as hitting someone with 30p on the head, same as 30b on the head. Ranged gets a higher bonus on head hits than melee, but that's because they are ranged weapons, not because they do pierce.
Blunt, Pierce and Cut only affects the armour calculation, whether a horse is reared or not (must be pierce is a requirement), and if a knockdown may happen (must be blunt is a requirement).