There is no difference between One-Click App and Droplet you setup manually. In both cases you get fully functional Droplet (VPS), which you can change it whatever way you want.

If you want to add one more site to your existing Droplet,
I recommend you to check out WordPress Multisite. It’ll allow you to use your existing installation to create more sites on same server.
You can learn more about it and how to do it in WordPress Multisite docs.

You can also install another WordPress in another location, create new database for it, but I think above way could be better.

In your case, @xMudrii probably has the best answer, but more generally, if you wish to rebuild a droplet and reset it back to its base distribution, I believe you should follow the instructions given at HowTo: rebuild.

In case that link disappears, the gist of the article is to login in to the DigitalOcean Website as the owner of the Droplet, and navigate to the Destroy panel. This gives you the option to rebuild the droplet from a limited number of distributions.

The documentation does say that this rebuilds it from the specified image and its original configuration parameters, so I am not 100% sure this removes the association with the 1-click application, or simply resets it.