Do you have "Download Originals" checked in the Mac's Photos Settings>iCloud? If so, then you have everything that's in iCloud.
If you have "Optimize Storage" checked, then you can not be sure how complete the Library is. Merging still works (I think) but if you have storage space, you should keep the Originals, even if it forces you to keep a Library on an external drive. Otherwise, a backup of the Library doesn't really backup up the full Library. To get a full Library, you make a new empty one, designate it as the System Library, connect it to iCloud, and let the empty Library fill up from iCloud. This could take days to weeks, depending on the size of the Library. Let me know if you need more details.
Speaking of external drives, how is the one you have formatted? You should know that 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
Is your "old backup on an external HD" an actual Photos Library, or is it straight files?