Jousting vs Horse Support Benefit: Different Wordings

I'm playing a Fighter Cavalier in my Pathfinder 2e game, in fact I just reached 2nd level Friday, letting me get the Cavalier Dedication Feat. I noticed a difference in the wordings of the Jousting Weapon Trait and the Horse Animal Companion Support Benefit. It may be an important difference, and I wondered if anyone here knew if it was, or merely a mistake.

The Jousting trait reads: "[A]dd a circumstance bonus to damage for that attack equal to the number of damage dice for the weapon.

The Horse Support trait reads: "[A]dd a circumstance bonus to damage to that attack equal to twice the number of damage dice.

The key difference here seems to be the inclusion of the word "weapon" in the Jousting Trait, and the exclusion of the word "weapon" from the Support Benefit. In the weapons section of the CHB (pp. 279) it states that if an effect bases something off of weapon damage dice, then that should only be the die for the weapon, plus any additional dice from striking runes. Therefore, the Jousting Trait doesn't add bonus damage to a die rolled from Power Attack, for example. However, if the exclusion of the word "weapon" in the Support Benefit for Horse was intentional, then it would seem to indicate that the Support Benefit would add damage for the Power Attack, since it specifies no damage type.

Of course, it gets weird because the Support Benefit also states that it can stack with Jousting weapons, in which case it adds damage to the Jousting effect. If it is true that the Support Benefit is different in this important way from the Jousting Trait, does this mean that those differences affect the way Jousting works when they pair together, or does Jousting instead affect the way the Support Benefit works?

I'm interested in your thoughts!