First thing: And how is your external hard drive formatted? To avoid damaging the Photos Library an external drive must be formatted in either APFS format or Mac OS Extended (Journaled) format. Additionally, the drive can not have had Time Machine on it since it was formatted. There have been so many problems with using incompatible drives that the newest macOSs won't even allow a Library on a non-Mac formatted drive to open, since there is a chance of damaging the Photos database. See this:
Move your Photos library to save space on your Mac - Apple Support
Are you using macOS 13 Ventura as your signature says?
A newer OS can't use an older Photos Library, but it can upgrade it. Usually if you try to open a Ventura Library in Sonoma, for instance, you'll be asked if you want to upgrade. This is important because an older OS can do absolutely nothing with a newer Photos Library. So if there's a chance you will want to use the Library on an older OS, you need to work with a copy.