"Late last year Amazon decided to discontinue the payments product that we have used."

I'm not necessarily saying you're wrong, I'm just not sure what else that could mean. When I see a button to pay with Amazon, the button says "Amazon Payments" on it, which I, as a layman, would call the Amazon Payments product if I had to call it anything.

Of course, Kickstarter may have just misspoke. Or, like you said, it's a specific product that only applied to Kickstarter. Either way, it's quite confusing from where I'm sitting.