IUP will give you an offer that you agree to at purchase. It's not a gift card, but a refund. The catch is, they don't refund you until their recycling partner gets the phone and inspects it. We had a user here who got really caught in that and FedEx or UPS lost his phone in transit. He had to wait several months to get his refund.
That said, his was the only story about that, that I came across, so it's rare. But to your question - it's a refund, not a gift card, but it's not instantaneous. I also don't think it will lower your monthly payments. It'll be one time refund back on your card - so if you have less balance that what they pay you, it'll just be a credit that sits there until later used to reconcile your balance. The loan on IUP is through Citizens Bank and not an installment plan on the Apple Card.