This has a lot of dependencies on your tech stack and what/how you want to pull your data.
In our implementation, I created an eVar for "direct source" that tracks "push-notification" or "universal-link" etc (for different direct to app entries). Our developers have code to detect this, and they pass me the value in a context variable on the first screen loaded in the app.
Specifically for our push notifications, I also have them set a context variable for "push id" on the first screen opened, so that we can correlate the open to not just "general push" but to a specific push that was triggered by the business team.
Then I also track the "push notification install Id" (on all sessions/pages regardless of coming from a push) which is tied to our push notification vendor, so that our data lake team can map out the Adobe users to the users in our push notification tool. This way we can better understand which users who are signed up for pushes are doing outside of the push, and IF they are actively using the pushes / % of visits that come from push or not / etc.
Some of this information make not be available to you.. you need to work with your app developers to see what they can and can't leverage from the pushes to support you.
If you are actually using the Adobe Launch pushes (and not a third party vendor like we are) then the integration may be easier.. you might be able to make rules right in Launch without a lot of developer interaction (this I can't really help with, as I've not used the Adobe pushes), but hopefully this gives you some ideas on what you might want to track, and how to go about it.
