@ggsgj6pvg0ex9y1 I think you're almost there, but definitely agree that the circuit board you have is completely different from Pogopooc's board. It looks to me that your board is being held on by four melted plastic pins that will probably need to be cut off in order to remove the board from the base. It's the same kind of mounting used for a lot of keyboards and other things where the manufacturer has assumed the part is never going to be removed; in other words, they assume it's non-repairable. Well, we've got news for them, 'cause we're going to fix it! :)
Here's your picture with those four plastic pins circled in red. Your two spring contacts (AKA pogo pins) are circled in yellow.
You can either take a utility knife and trim those four pins off, or a soldering iron will also make short work of them. Once they're off, you should be able to take the circuit board off the base. From there you'll need to figure out what kind of replacement pins you need. The fact that you have solder domes on the reverse side of the board tells me you're looking for what are called through hole pins rather than the surface mount pins used on Pogopooc's board and the Ember mug charging base that I'm working on right now.
In general you'll want to measure the length of the pins and use that as your main search criteria for finding replacements as well as looking for through hole rather than surface mount too.