How does SegWit reduce transaction size, when the signature is simply moved to another part of the transaction?

How does SegWit reduce transaction size, when the signature is simply moved to another part of the transaction?

This video explains that the signature is moved out of the input section and moved to the new one called the witness. From what I can tell, the witness signature is still contained in this transaction - how would a simple reordering of information magically create more space?

I've perused other articles and videos and none of them seem to address this "magic".

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