In iOS 17.2 we can start a live activity by push, and the document says:
When the system receives the ActivityKit push notification on a device, it starts a new Live Activity, wakes up your app, and grants it background run time to allow you to download assets that the Live Activity needs.
My problem is, in which callback can I be notified a live activity will be start by a push, so I can download some web contents such as images before the live activity be rendered.
Your Live Activity process, in most cases, will be woken up by the system when a new token arrives. This would be the time to get any needed contents (or earlier) as opposed to when sending the push, because as you've mentioned, there is no callback for a Live Activity push. More specifically the callbacks for pushToStartToken and pushToken.