My opponent played a Seal of Discord, then he played an Astral Heron to a battlefield he controls. Does he get to trigger the Astral Heron ability discount for his next card played? My opponent's argument is the Astral Heron went on the battlefield and will count himself as the first card played when he is on the battlefield, thus fulfilling the condition. The battlefield is a binding condition.

No — the discount never fires. Seal of Discord was the first card played that turn, and it resolved before Astral Heron was anywhere on the board, so the "first card" window was spent with no Heron to see it. Heron's trigger is checked when that first card finishes resolving (CR 420.4.a), so playing him second just leaves him sitting there.

Your opponent is right about the battlefield half of the condition — if Heron is genuinely the first card played and resolves straight to a battlefield, he sees his own play and the discount applies (FAQ #13157). But "first card each turn" is a one-shot window, not something a later Heron can retroactively reclaim, which is exactly FAQ #12289: play a card, then play Heron to a battlefield, no discount. The battlefield clause being "binding" doesn't matter here; it was simply never reached.

So his next card pays full price. Worth noting the trigger would have worked if he'd played Heron first and the Seal second — and that discount expires at end of turn if unused.