From the perspective of an orthodox boxer...
It is very important to keep your shoulders bladed i.e. left extended, right shoulder back. There are many passive defensive benefits that this stance grants you. Your left shoulder becomes a shield from right-handed attacks, and is especially effective to tuck your chin behind when pivoting clockwise away from an orthodox opponent's right hand.
From this stance, your right hand is given a more clearly defined purpose for defense. You can choose to position it on the right side of your face (more traditional) to shut down the left hook, or just can place it in front of your face to shut down one jabbing lane for your opponent. If you choose to defend the jabbing lane, you must be aware that there is an opening on your defensive radar for an incoming left hook. However, it is a VERY easy and compact movement to roll your shoulders as if you're throwing a cross and slide your right hand up high to catch the hook. This also sets up a loaded left hook counter for you, or you just shift your shoulder positioning back to normal and get back to work. You're not as open to the hook as you think from here, however... using this type of shoulder positioning puts your chin a few precious inches further away which means they either have to reach more for that hook, or commit their feet to it and really close range.
Fun trick associated with the last point on right hand positioning:
If you prefer to keep your hand in front of the face to close their jab lane as I often do, it can set you up very nicely to counter the jab. When you are ready to counter your opponent's jab, just move your hand back to the 'traditional' spot beside your face. After frustrating your opponent with a closed jabbing lane, they will find it very hard to resist this opening. Once you move that right hand, anticipate their jab and act accordingly... whether it is a parry into a counter jab, a slip into a counter, whatever. Just know that it is YOUR jabbing lane, and you get to choose when it is open.
This type of shoulder positioning opens an entire range of "shoulder play" which sets up different counters and active defenses. Offensively, it can also create some opportunities. If you keep the elbows in in conjuction with a leading left shoulder, you can discreetly line your jabbing hand up inside of their guard, inside of their punches, etc from the relative safety your hand positioning grants you. From here, drive off of the back foot and emphasize a slight turn of the upper body for the jab. The arm is completely taken out of the equation, and what you get is a ruler-straight jab that you can pop into any holes you see. I don't think I'm articulating this well!
Also, Sinister, while I appreciate the emphasis on the basics in your drilling video, your boy is jabbing directly at your glove and his partners glove!