The answer is no, the creature is not provoking an attack of opportunity--Seize the Moment is merely providing an AoO for something you normally wouldn't get. Contrast this with the wording of Outflank (taken from pfsrd)
Outflank (Combat, Teamwork) You look for every edge when flanking an
enemy.
Prerequisite: Base attack bonus +4.
Benefit: Whenever you and an ally who also has this feat are flanking
the same creature, your flanking bonus on attack rolls increases to
+4. In addition, whenever you score a critical hit against the flanked creature, it provokes an attack of opportunity from your ally.
The wording of these two feats is very different, and you can see via the italicized portion that if seize the moment were intended to work the way your friend describes it would also have to be worded this way.
First of all, I always assumed Bonus=Benefit in the context of solo tactics. Anything else didn't even occur to me.
Regarding your question, the FAQ offers a precedent for Benefit = Bonus in the context of Outflank's secondary ability.
Benefit: [...] In addition, whenever you score a critical hit against the flanked creature, it provokes an attack of opportunity from your ally.
An attack of opportunity is most definitely no Bonus in the sense of the rules. However, the FAQ states, with regard to exactly this part:
Inquisitor: If an inquisitor uses Solo Tactics (Advanced Player's Guide, page 40) with the Outflank feat (APG, page 165), does the enemy provoke attacks of opportunity when hit with a critical hit?
Yes, but only when the inquisitors allies score a critical hit against a foe that they both flank. In this case, the enemy provokes an attack of opportunity from the inquisitor. The reverse is not true, since her allies can only gain bonuses from teamwork feats if they themselves possess them.
So for the purpose of Solo Tactics, "getting an AoO" is considered a Bonus, however "granting your ally an AoO" is not, which to be honest, is a little odd, since the Benefit of the feat clearly describes the latter.
I think the general rule with Solo Tactics can be expressed as
"Anytime it would be good for you if one of your allies had the same Teamwork feat as you, you can pretend he has it."
This is compatible with the FAQ ruling and may or may not be RAI (We might never find out). Some specific Examples as to what this reading means:
- Outflank: "If your flanking buddy ally lands a critical hit,
if he had Outflank, you could make an AoO."
- Coordinated Reposition: "If your ally makes a 5 foot step,
if he had Coordinated Reposition, you could also make one as an immediate action"
- Escape Route: "If your ally threatens a square,
if he had Escape Route, you could walk right through there without provoking an AoO."
Best Answer
Maybe? Ask the GM
This is an area where the game explicitly relies on the GM's judgment. There's no way to know which way a particular GM will rule until the player asks.
As cavaliers of the Order of the Dragon
…that description sounds to this GM as if such a cavalier should probably get the bonus himself. I can see another GM confronting me down at Hogwaller's for allowing that to work that way and that other GM saying that it "makes no sense" for an Order of the Dragon cavalier's challenge to benefit the cavalier because he's all about helping not himself but others. I'd counter with killing bad guys is helping others, even if the cavalier's alone. Punches would be thrown, and we'd spend the night in jail. After we're released, he'd challenge me to a PvP duel to settle things. I'd win, but, while we were dueling, his buddies would key my car.
Anyway, ultimately this GM figures the bonus is so little that it won't matter in the long run and would allow the cavalier to benefit from his own challenge, but a less generous GM might not. Steel yourself for disappointment. Or save up for some auto detailing.
(By the way, Paizo messageboard threads from 2014 and 2015 generally agree the Order of the Dragon cavalier should benefit from the challenge himself.)