The choice is up to you; either way will work. Some considerations:
1. If you put the catalog on the external drive with the images, LR is likely to be slower. Putting the catalog on an internal drive is a speed benefit.
2. If you put the catalog on the external drive, you can move the drive to another computer running LR and open the catalog and use it on the other computer. You’ll have a portable catalog solution. However, you may need to reconnect your folders if the drive letter or name changes when you plug it into another system.
Thank you. That's helpful.
You will be able to work with Lr, when your external drive with the catalog on the external drive is NOT connected to your computer, but you will not have access to your catalog and data.
If the catalog is internal, the prictures are external, you will be able to edit offline, because Lr keeps a cache of the cataliog images and for most operations that‘s enough. Buit you will need to connect rthe externl drive for exports in full resolution. And you cannot connect your external drive to a second computer and expect to be able to continue your edits there.
And: to be secured, you need to backup both: the pictures AND the catalog and this for all solutions you choose.
You need to know: the images are as they are and they will stay untouched. The catalog contains the edits your doing and (resolution reduced) previews of the images. By exporting an image, you are applying youir edits to the original image and saving that as a new image.