iCloud Photos not syncing all photos to Mac

Since the iPhone and iCloud now show about the same number, I’d first check the Mac’s actual Photos library.

With the external library open, go to Photos > Settings > General. If Use as System Photo Library is grayed out, that library is already the System Photo Library. That matters because only the System Photo Library syncs with iCloud Photos.

Then go to Photos > Settings > iCloud and confirm iCloud Photos and Download Originals to this Mac are enabled. In Photos, scroll to the bottom of the Library view and check the sync status. It may simply still be downloading from iCloud.

One important point: iCloud Photos is synchronization, not a separate backup. Deletions also sync, so I wouldn’t delete, move, or create another library while sorting this out. Apple recommends keeping a separate backup of the Photos library even when using iCloud Photos.

Like PreCognition, I'm concerned that you say, "my library…is located on an external drive to keep a back up to iCloud." iCloud Photos is not a backup service. See this:

Understanding iCloud Photos - Apple Community

Also, how is your external drive formatted? To avoid damaging the Photos Library an external drive must be formatted in either APFS format or Mac OS Extended (Journaled) format. The drive must be connected directly to the Mac by cable, not networked, clouded, NASed, etc. Additionally, the drive can not have had Time Machine on it since it was formatted. See this:

Move your Photos library to save space on your Mac           

If this drive is in a an incompatible format, stop running Photos with it immediately!  A Photos Library can sit on an incompatible drive, but running it may corrupt the database.

On a Mac it's possible to have more than one Photos Library. You can do a Finder search for ".photoslibrary" to see if you have others. Or, you can close Photos and option click on the Photos icon to get a list of Libraries that Photos knows about.

Let us know what you find…