Yes. I don’t dispute any of that. But nothing in that text says you take the Magic action. In fact, it wouldn’t make sense for you to do so, because you only have one action per turn, which you’re already using to take the Ready action.
The different named action types, much like turns, are strictly meta-game rules constructs. Taking the Magic action is not something you do as part of “casting a spell normally,” it’s just the rule that most typically allows you to “cast a spell normally.” The Ready action also allows you to cast a spell normally, and “release” it as a Reaction in response to a later trigger.