Photos in External Hard drive missing between computers

I ran out of space on my computer (and iCloud) for all my photos so I bought an external hard drive and moved the photo library to that drive. (Now I can only open the photos app if the hard drive is connected). However, a new problem has shown up now that I have a newer computer that runs on a different operating system. It went like this:

1. Seeing photos from old 2013 computer on new 2020 computer via external hard drive transfer: it worked fine, though I did get a pop up where it said they had to change the library thanks to the updated system on my 2020 computer, which was fine.
2. Moving back to the old computer to load more photos onto the external hard drive: Where things got weird. I had to load the photos onto the old computer first because new Macs don't have an SD card slot (yippee USB-c!), and when I opened Photos on the old computer, it was unhappy because Photos was too advanced for the old thing, and it created a new photo library (Library 2). This was fine, except now I can't find Library 2. Anywhere. This is unfortunate, as I like to delete photos from my SD card once I load them onto a computer to save space (which maybe this will teach me a lesson about that), so now I'm missing a whole photoshoot and a fair amount of more "documentary" photos of my family that I'd really like to have back. Any help as to where in the world a second library would be hiding on the hard drive would be very nice. Thank you!