Notification Center UX

I’m working with a client on their notification center for their web app.

They have a business need to send mobile push notifications and also in-app notifications/alert messages for the same goal but with slightly different text (shorter text for the mobile push) AND send them on different dates. However, when the user goes to the notification center, they could either:

See the list of notifications with the two messages related to the same action with different dates (one sent out as a mobile push and the other within the app) - this seems odd and redundant to me.

See the most recently sent message replace the older (so the mobile push was sent first, then the in-app notification sent second, so once the in-app is received the user no longer views the mobile push notification).

Separate the messages out in two tabs in the notification center one for the in-app alerts and the other for the mobile push notifications. Something like the following:

I haven’t found much on this use case, so any insight would be much appreciated.