I only want to monitor certain backlinks that I specify - can I do that?

Yes, you can certainly do that.

You should:

Add the domain (you should get guided through this automatically during your signup process. If it didn't work, you can also refer to this article.)
Disable auto discover.
Add your links manually

Please note that an alternative that you should consider is to:
 - keep auto discover enabled (to continue seeing all your links)
 - add tags for the links you import manually

This way, you can get the best of both worlds (being able to see all your links, while also being able to filter specifically for the links that are especially important for you.)

Updated on: 24/08/2019