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This article from Amnesty International's Security Lab is a good primer on the attack vector used by the Pegasus exploit

https://www.amnesty.org/en/latest/research/2021/07/forensic-...

From my understanding, the attack can happen through multiple ways. Companies like NSO purchase vulnerabilities for this step and probably have a set of them on their disposal, ready to deploy ones as they get discovered. Pegasus is the rootkit which is installed once the attack is successful, and allows spying on the user. https://en.wikipedia.org/wiki/Pegasus_(spyware)

The iMessage exploit sounds like a Null terminated string issue.

The rest of it sounds like a Metasploit-esque framework along with some telemetry tools.

Pegasus is a name for their attack repertoire of the day plus whatever next stage payloads are run upon a successful compromise.

iOS iMessage bugs parsing images, I recall

From the Amnesty report shown by another commentator, there were a bunch of compromised Apple services.