The help topic you post is not about moving your library - it is about using a referenced library - which is strongly not recommended and since you are confused you should not do it (IMHO even if you are very well informed you should not)

Moving the library is very simple

You need a directly  connected drive formatted Mac OS extended (journaled) (not sure where you see conflicting advice on that but that is the format you must have) and you quit photos and drag the Photos library to the EHD then hold down the option key while launching Photos and select the new location - make sure your backup program is backing up the EHD and after fully testing the move trash the library on the internal drive (test one more time before emptying the trash)

LN

Please do not ThreadkJack other people's threads - that is Rude and very confusing to you, the OP and the volunteers here - when you have a new question please start a new thread - you now have posted two totally off subject posts in this thread

LN

When I take pics with my phone.. do they properly go into the Library.. are they "references files" that need to be manually added.. or will they just add into my iCloud library automatically?

That will depend on your settings for iCloud Photo Library on your iPhone and your Mac.

• If iCloud Photo Library is enabled on your iPhone, the original photos you take on your iPhone will upload to iCloud Photo Library.  And when you enable iCloud Photo Library on your Mac as well, the photos will automatically download to your Mac as managed photos, stored inside your Photos Library.
• Or if you use My Photo stream instead of iCloud Photo Library, the photos will also stream to your Photos Library and be imported as managed.

Will I still be able to browse my photos on my MacBook Air when disconnected from the external drive or will it be complicated some how?

If you use referenced photos and the drive with referenced photos is offline, you can browse the thumbnails, but not edit the photos or print or export them.

My library is clearly too large to reside on my computer so I have to move to external drive.. just wanting to know caveats.

Copy the complete library to the external drive and then connect the drive when you want to work with your photos.

Also.. I've seen conflicting information on various sites about how exactly the external drive needs to be formatted.

If you want to use iCloud with your Photos library, format the drive MacOS Extended (Journaled).  iCloud Photo Library will not work otherwise.  And you cannot use iCloud Photo Library, if you import the photos as referenced.

Wait with deleting the old iPhoto Library. You will not gain much space by deleting it, because the new Photos Library and the iphoto Library are sharing the photos inside using hard links. Deleting the iPhoto Library will not free much space, unless you delete the photos from Photos as well.

See:  Photos saves disk space by sharing images with your iPhoto or Aperture libraries - Apple Support,  also Six Colors: The (hard) link between Photos and iPhoto

You may need the iPhoto Library to fix problems in the Photos library by repeating the migration.  At least keep a backup copy.

If you move the Photos library to the Lacie the hard links will be broken anyway.  Then move the iPhoto Library as well, if you need to free space.

It looks like the community hosts kindly branched the question to a new discussion. I did not see any thread jacking when I answered. 

It is possible - that is called iCloud Photo Library and does not involve a local hard drive - your library is on iCloud

What you did is totally different and requires that the hard drive be available anytime you launch Photos

LN

David, you need access to the Photos Library to be able to browse your Photos.

If the Photos library is on the external drive, the drive needs to be connected.

How much storage has the internal drive of your MacBook Air?

You have essentially three options to keep the library on the internal drive to be able to browse your photos with the drive offline:

1. Split the library:  Keep your most recent photos in a separate library on the internal drive and most of the older photos on the external drive.
2. iCloud Photo Library:  Sign up for enough iCloud Storage to keep all original files in iCloud and use the "Optimise Mac Storage" option. Then you can upload the Photos library to iCloud, and download it again from iCloud to your internal drive.  You will need about 20% of the storage for the full size of the library this way. You will need an internet connection to be able to edit the photos or browse them at full size, and the additional cloud storage is not free,
3. A referenced library: As discussed at the start  of the thread.  Keep the library on the internal drive, but reference the original image files on the external. It is essential the same scheme as the iCloud Library option. The original files are stored elsewhere. But as Larry pointed out, this is not recommended, because referenced images are not well supported.  And you cannot turn an existing Photos Library into a referenced library. You can make a photo referenced, when you import it, when you start with a new library.But managed photos cannot be converted to referenced photos. You would have o export them and reimport them.

what are your setting for iCloud in the system preferences on your Mac and in Photos on your Mac and on each IOS device?

LN

and my whole library is in iCloud.

Then you should be able to view the library right now already, even with your drive offline from the web interface at  https://www.icloud.com/

Open this page, sign on with your Apple ID, click the "Photos" button.  You can view all photos, upload or download photos, but not edit them.