I deleted facebook with a desire to reinstall it, but in my settings FB appears as "installed", can't install it through AppStore because my device still thinks it is already installed.

It's possible but you should be able to delete an app regardless. It's always recommended to update your device to the latest iOS.

You could try a hard reset. Hold the Power and Home buttons at the same time until the phone turns off. I have found this to clear things up better than just powering the phone off.

And just to be sure, swipe down on your screen and search for Facebook. If the top hit is the Facebook app, then it's not deleted from the phone. If the top hit is something like a link to Facebook, or something from your browsing history, then the app is not on the phone. Do you physically see the Facebook app still or does it only show up in Settings?

Restart the phone and try again.

Also, are you deleting it from Settings > General > Storage & iCloud Usage, under the Storage section? If you are deleting it from Settings > General > Storage & iCloud Usage, under the iCloud section, you are only deleting the data associated with Facebook.

Do the update and see if it helps. Syncing with iTunes can also clear up invalid storage information.