Do you have a backup copy of the library, made before you tried to open the library on the Mac with the newer version of Photos? It is not possible to share a library between two Macs with a different version of Photos. As soon as the newer Photos version opens the library, it will be converted to a format, that the older Mac cannot read.
Try to get the new MacBook Air to fix the library. Opening and preparing a large library with 400GB of photos and videos can take several hours.
- Make sure the external volume has the "Ignore ownership on this volume" flag set. This is described here: Move your Photos library to save space on your Mac - Apple Support. This flag will help to avoid permission and ownership problems, when working from different user accounts on different Macs.
- Make sure, the library is on volume, that has not been used for Time machine backups, or you cannot work with the library on Catalina Macs.
Once the volume has been prepared, try again to open the library in Photos. Let it run, even if it takes hours.
One problem may be incompatible media files in the library. Photos on Catalina cannot work with all media formats that may have been imported to the library on older system versions. It would have been better to remove files that can no longer be used on Catalina from the library before opening it in Photos 5.
This list (for iMovie) can give you an idea what to look for:
About incompatible media in iMovie for macOS - Apple Support
