I don't want to damage the door by prying, so I'm hoping there's some other reasonable option. This was addressed before, but my hinge is different.
Turns out that wedging a thin blade screwdriver under the door mount (as shown by the three horizontal arrows) was the way to release these old hinges. They were of the sort that were hammered in and instead of screw holes underneath the cover, there were two 1/4"x3/8" knobs that went into holes beneath the cover. Seems like pretty poor design, and a lot more work for the cabinet builder, but these were made over 20 years ago, so perhaps that's how it was done then. Hopefully the replacements from Amazon will work as well, since that particular door that conceals the wastebasket is opened at least 5x more than any other door in the kitchen.