Aircraft Carrier Launches: ABE Checks and Signals
I'll take a stab at this.  It's been many years, but I was a shooter (yellow vest, green helmet).  Someone with a better memory can chime in.

You're correct that one of the hook-up operators is under instruction (U/I).  There are also 2 shooters in this video; one of them is under instruction as well.  Sometimes the instructor will give a thumbs up to indicate that the U/I operator's signals are correct and should be obeyed.


Before 0:27 they are looking toward the "island" for a green beacon that indicates the Air Boss is ready for the crew to launch planes.  At 0:27 they see the beacon turn green, then they look to the shooter for the hook-up signal.  The shooter is checking for several things: green beacon on, deck clear forward, jet blast deflector (JBD) raised, nobody on the wrong side of the foul-deck line, proper aircraft configuration (flaps down, tailhook up, etc), wind speed hasn't dropped since the catapult steam setting was entered, and thumbs up from the squadron QA folks (white shirts, behind aircraft on both sides) who are watching the jet for proper functioning, no leaks, etc.  When all those conditions are met, the shooter gives the hook-up signal, which triggers the hook-up operator to call for the jet to taxi forward.  The yellow-shirt (ABH) receives the "rolling" signal from the hook-up operator, then signals the pilot to pull forward, which causes the tow bar to drop down in front of the shuttle.
I think they are checking for a thumbs-up from the center-deck operator (sitting in the hole next to the shooter; thumbs up indicates good catapult steam pressure and good wind over the deck, which the shooter is no longer monitoring).  They also may be looking at some of the same things the shooter is looking at. 
The holdback bar needs to be properly set in its slot in the deck, properly attached to the aircraft, not leaking hydraulic fluid, and IIRC there's a shiny metal ring between the orange and white sections that indicates proper functioning and tension.
The deck edge turns his whole body so everyone can see that he's looking fore and aft for clear deck forward and JBD up, and maybe also so he can see past his arms which are raised so that everyone can see he's not pushing the launch button yet.  At this point the shooter hasn't been looking forward because he has been mainly focused on the pilot and aircraft while the engine is running up and the pilot goes through flight-control checks, salute, etc.


Additional comments: most of the time the shooter and center-deck operator are below decks inside the glass "bubble" but they do practice deck launches to stay proficient.  I did my shooting on an older ship that didn't have the bubble at all.  Needless to say, communication is purely visual because you can't hear anything.  Even the operators with radio or phone headsets can't hear anything when jet engines are blasting nearby.

Finally, if you think this looks complicated and dangerous, remember this video was taken in daytime in good weather.