That is indeed the way webCoRE differentiates between sending push notifications and also ‘storing them to messages’. To just ‘send a notification’ it uses
sendNotificationEvent(). Notify Me When uses
sendNotification(), letting it default to also sending the notification event (the same thing as storing as a message).
It seems like the new app is only working with notification events, and has no interest in the legacy push notifications.
Something you might like to try, but equally might be a pointless suggestion, is to look at your notifications via the REST API. In Curl command line speak it is:
curl --header "Authorization: Bearer <Personal Access Token>" https://api.smartthings.com/notification
(visiting https://account.smartthings.com/tokens for a token with Notifications scope).
It doesn’t seem impossible that the notifications might be in the feed but not get acted on by your app.