It's not needed to publish the app to install it on your store. Without installation also you can do it.
Store-name:- Use your store name
api_key:- Use your's app api key to install on your store.
As this is private app so have you created this app for the same store where you are installing it, because as per the screenshot provided it's already giving the alert that "This app is created for one store and you are installing it on another"
So first let me know about this inf so that I can suggest you further.