Upon a customer’s initial subscription, whether with or without an offer, you will receive the SUBSCRIBED notificationType. If a customer is currently in an offer period, their conversion is equivalent to a renewal, thus you would receive the DID_RENEW notificationType.
For further reference, I recommend exploring the documentation provided on all the notification types sent and use cases listed at the bottom:
https://developer.apple.com/documentation/appstoreservernotifications/notificationtype
Additionally, at WWDC24, Alex delivered an informative talk on notifications that may prove beneficial to you. You can access it via the following link:
https://developer.apple.com/videos/play/wwdc2024/10062/?time=271