I think you confusing something, the video shows how you can take off the Anti-Shatter Screen Protector (ASF) which come with the phone. If you look closely to the glass of you phone you will see there still a tiny screen protector and if you remove it the Sony logo will go out with him because the logo is on it. So if you apply any other screen protector on the phone the Sony logo will not come off if one day presumably you will try to remove the screen protector which you apply. BTW is not recommended to remove the original anti-Satter screen protector from Xperia device.
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My Evolution: Sony Xperia Z1, HTC HD2, HTC Diamond 2, HTC HD Blackstone, Nokia 7070 Prism, Qtek 9100, Nokia 3100, Motorola T720i, Nokia 3310, Sony Ericson PF 768 (Blue)