It's also a stretch to think "normal" = Ready Action, because if it did the phrase would never have been used.
The "normal" you are quoting in fact comes from the description of the Ready action in the rules glossary. That use of "normal" you are quoting is specific to the Ready action.
The phrase "When you Ready a spell, you cast it as normal..." only appears in the description of the Ready action. It appears nowhere else in the PHB AFAIK.
As such, absolutely, the phrase above "When you Ready a spell, you cast it as normal" == Ready action (ready a spell specifically)
Not only is that not a stretch, it is specifically what it is referring to (whether or not you also use a Magic action as part of that Ready action).
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