If Astral Heron is at a battlefield, a card is played, and then Astral Heron dies, does the cost reduction trigger still stay around for the next spell?

Yes. Once Astral Heron's trigger condition is met (you played your first card while it was at a battlefield), the ability goes on the Chain and resolves independently of the Heron — its death doesn't cancel the discount.

The "if I'm at a battlefield" check happens at the moment the trigger condition is evaluated, not when the ability resolves (Rule 383.2.c). A Heron that dies afterward — even in reaction to the trigger — still generates "[2][A][A] off your next card this turn," because the discount is a pending effect that doesn't require the Heron to remain on board (FAQ #11683, #11729). Same as the bounce case: removal after the trigger fires changes nothing.