Whelp, the news about these batteries is good and bad. Both batteries listed below will run both Cosmo and Gemini for 3-5 hours, depending on use. A battery that has this lettering:
https://www.ebay.com/itm/203189521885works with my Gemini, both running and charging.
And battery that has this lettering:
https://www.ebay.com/itm/175240435655successfully runs in the Cosmo, but won't charge (nor will the first-listed battery charge in the Cosmo). My results are the opposite of Davide's (blogger in first thread post). Though he doesn't own the first-listed battery, he can get the second-listed battery to run/charge in his Cosmo, but he can't get it to charge in his Gemini.
At first, I thought my Cosmo's charging was broken, but it can still charge the original battery, so I'm very confused. I made sure to buy Davide's battery lettering for the Cosmo, but no go.
Since I no longer use PC devices for anything except wifi-ed Gemian, I'm at least glad to have the Gemini up and running, and for a nominal fee.
Jake