How to use desktop and laptop in LR?

What are the specific on that external hard drive?Make/model/type.

It is a hard drive and not flash media such as a thumb device, correct?

Method 1

1. Open Lightroom on your computer. You’ll find it in the All Apps area of the Start menu (Windows) or the Applications folder (macOS).

2. Connect the camera or drive to your computer. If the photos are on a camera or removable storage card, connect the card reader to the computer.

3. Click +. It’s at the top-left corner of Lightroom. A context menu will appear, displaying all connected devices.

4. Select the device where your photos are stored. If the photos are on your computer, click Browse… to open your computer’s file browser.[1]

5. Select the photos you want to add. Clicking a photo preview adds a check mark to the photo, indicating that it’s selected.
If the photos are all in one folder, select the folder, then click Review for Import (Mac) or Choose Folder (PC).
To select one or more individual photos, click each photo you want to add (hold Ctrl (PC) or ⌘ Command (Mac) as you click to select multiple photos at once), then click Review for Import.
Image titled Use Lightroom on PC or Mac Step 6
6. Click Add Photos. The selected photos are now added to Lightroom.
Method 2
Creating Albums and Adding Photos

1. Open Lightroom. You’ll find it in the All Apps area of the Start menu (Windows) or the Applications folder (macOS).

2. Click Add an Album. It’s in the left panel under the Albums header.[2]

3. Type an album name and click Create. This creates the album and adds it to the "Albums" section of the left panel.

4. Click the album name. This opens the album, which currently contains no photos.

5. Drag photos from the grid to the album. Another way to add photos is to click the blue Select Files… button, then select the photos you want to add.

Greeting,

Rachel Gomez

You can't run the desktop and laptop at the same time. You can have both installed, of course, and you can store the catalog on removable media; this issue may be due to the Mac and how Privacy is set there. In System Preferences, >Security & Privacy, >Full Disk Access, do you see Lightroom listed? If not, do add it and try again.
See: https://community.adobe.com/t5/lightroom-classic-discussions/quick-tips-how-to-give-full-disk-access-to-lightroom-classic-on-macos/td-p/12913413