Moving back-up iPhoto to Mac

There's something peculiar here. Time Machine copies files but never moves any, so you have to ask yourself how that Library got moved. It wasn't TM.

This link

Restore a library from Time Machine in Photos on Mac – Apple Support (UK)

explains how to restore a library from TM.

I want to help, but you didn't answer a number of important questions. Also, you said you are not using iPhoto, but, you said, "The photos icon has Photo.app written underneath it. " If that's correct, then this is not an Apple product. Apple's replacement for iPhoto would be "Photos.app." We seem to keep getting off by one letter, and for this stuff that could be significant. I'm guessing you mean Photos, but that must be straightened out. You should see the Photos.app in your "Applications" folder. And could you answer this? Is the "Photos file" you mentioned labeled with a ".iphotoslibrary" suffix?

Do you see a file with the ".iphotoslibrary" suffix somewhere on the external drive? Do you see such a file in the Pictures folder on your internal drive? Can you open Photos at all? What's there? What happens if you double click on something with the ".iphotoslibrary" suffix?

You said you used Time Machine for a backup. Are the Time Machine files on this external drive? How is your external drive formatted? You may want to refer to this:

Move your Photos library to save space on your Mac - Apple Support

Once we get what you have straight, we can make some progress!

You can certainly use your external drive s is to open, explore, and order photos using photos. Otherwise, just drag your library back from the hard drive to your Mac. Provided that your photos are included in photos and not referenced, everything will work

You have not said what macOS you are using-- please let us know. Also, you are referring to iPhoto, a program that was discontinued 8 years ago-- Are you still using iPhoto, and is the Library labeled with an ".iphotoslibrary" suffix? You talk about a "Photos file:" what do you mean by this? What is the suffix on this Photos file, and how much storage does it use?

Time Machine doesn't move files; it makes copies.

Since the pictures are on you other devices, you can get them back to the Mac. But we need to be sure of these files to be sure just how to do it. Let us know-- all will be good.

And whoops-- I didn't see the most recent exchange-- sometimes this thing is really slow!