What to do when caught inside?

[sflinux]

Yes, pushing down on the tail is key. You put your back to the whitewater, point the board toward shore, and the last second before the wave hits, you push down on the tail. The nose of the board will point up and the wave should pass by without taking it. Having a kick pad helps. It works best with smaller to medium-sized boards.
I hold the paddle with one hand put one hand on top of the other as I push down.

I use a variation of this technique.  I think my version stems from kitesurfing, where you have one hand on the bar, and you can want to grab onto the board before a wave hits it, or else it gets carried to shore (leashless).
Face the beach, point the nose of the board toward shore, with my backhand I reach over to the far side of the board so my bent elbow will be near the kick pad, I pull the board to my chest and squeeze.  The last second before the wave hits, I push down on my elbow (use your body weight too) to push down on the tail.  If done properly, the nose of the board will point up and the wave should pass by without taking it.  With SUP, instead of a bar, I have a paddle.  If there is enough time, I lay the paddle on the board, keep a grip on the paddle with my front hand and focus on the tail of the board.  The main hazard of this technique is the fins, but seems to work well in waves under head high.  Same as a handle, if something feels sketchy when the wave hits, let go of the board.  Maybe worth mentioning, I wear an impact vest.

One thing I like to do whenever possible is turn the board nose first to the white water or breaking waves and give the nose a side hug. Gripping the board as hard as I can under my arm. Just before the wave hits, sink the nose and yourself as deep as possible, basically trying to get the tail out of the water.

I use this technique too (borrowed from bodyboarding).  The smaller the board (surface area/volume), the better this technique works.  Works well in small and soft waves.  I don't have a death grip on the board, if something feels sketchy when the wave hits, I let go of the board.  Maybe worth mentioning, the paddle is in the opposite hand.  Best not to hold the handle near the middle of the shaft (near the handle is preferred), because the blade will act like a lever when the wave hits it (I hurt my thumb once).  I keep a loose grip on the paddle (if the wave wants to rotate the paddle, let it), and if something feels sketchy when the wave hits the paddle, I will let go. 
In bigger surf with long hold downs (letting board go to leash, preferrably a waist leash), hand starts near handle but after turbulance and while underwater I slide my hands to the blade and using the blade like a hand plane to give me more paddle power to get to the surface.

[OkiWild]

I've used Dakine leases since they've bees selling them. Most recently with SUP, I use the Kainui 5/16" x 9' leashes.

I inspect my leashes religiously before going out, and after any wipeout where I suspect a fin nick. The Dakine leashes are pretty much bullet proof, but have one weak area that needs to be inspected. At the rail saver, where it attaches to the stainless swivel. The rope inside will eventually wear, and the white, inner cords start to show. At this point, the attach point is on very borrowed time. Like if you find white, don't use it. The more the board spends at the end of the leash on a hard pull, the faster this wears. I've never had one wear out in less than a year, though.

Attaching pics of a bad and good rail saver swivel rope.

[exiled]

These are great photos OkiWild. This is the point that all of my big wave DaKine leashes have failed at, usually at around 12-18 months. The Leashlok design is far superior in this regard. My first Leashlok is still going after almost 5 years, but given the nature of urethane it has to go eventually.

[surfinJ]

https://staycovered.com/collections/big-wave-surf-leash

This company produces high quality handtied goods. My 3rd winter of some longperiod bashings and no broken leashes yet.

[jondrums]

I'm having really good luck with the XM .313" Hand-Tied leashes.  Reasonably enough priced, ships fast, 100% made in the US including the cord.