I'm actually on the same path. The seller for the battery you linked to seems to be using the same marketing info as this product: https://www.amazon.com/Playstation-Replacement-CUH-ZCT2U-Bluetooth-Controller/dp/B07YDR9KRG

I bought one the other day and am getting a spreadsheet generated to test the discharge rate. So far, it seems to be no different from the original 1560mAh battery, so I'm going to have to do something different to make sure this is all correct. Maybe there's a better way to test than just leaving the controller on for 12 hours, but at the moment it's behaving just the same as the old battery. Kind of a shame, really.

If you want, I can host the data somewhere once it's all complete if you want to take a look at what I find.