Q: how do i install a purchased app that says installed but it's not on mac or iPad?

Follow the instructions here > Downloading past purchases from the App Store, iBookstore, and iTunes Store

You must use the same Apple ID used for the original download.

You are mixing two different types of apps. Mac apps don't run on iOS devices like an iPad. And iOS apps for iPads don't run on Macs. These two different devices; the Mac & iPad, would need their own version of an app for both to have the app.

An app is considered to be installed if it shows up in a Spotlight search of local volumes. Bring up a search window in the Finder by pressing the key combination command-F. Search This Mac for the name of the application. Delete all copies that are found.