Thanks MT ;)

The site I referenced as you did also, is for the Lance Armstrong Live Strong Foundation.
He's famous for being a biker, but like all world class athletes, he cross trains. I know he runs/lifts weights, and wouldn't be surprised if he did some Yoga or Pilate's as well.

The picture was meant to convey nose breathing during strenuous activity. I'm guessing certain yoga poses would be considered strenuous at least for me they would be ;D

I can see how you might think it was a Yoga site, but if you read all of the info, it's clear that it's not. MT"s link leaves little doubt that the experts they refernece believe you should breath in thru nose, out thru mouth.
 
They mentioned some yoga poses that dictate you breath thru your mouth, but that's for specific working of core muscles, a technique like I believe your mouth breathing when climbing is. They also reference the fact that mouth breathing stimulates the flight or flight response. Tenses the body for the conflict to come. That tenseness may be useful for working deep core muscles and probably some martial arts techniques.

I'm a big fan of boxing/mixed martial arts fights. Ever see a boxer using nasal decongestants before the fight to clear their nasal airways.
Ever hear the fight announcers/analysts, comment that he's " sucking air", breathing heavily thru the mouth? That's a sure sign that their out of gas and ripe for the taking. Same with Basketball players on the court during foul shots. If they have their hands on their knees and breathing deeply thru mouth, it's sure sign their gassed.

If anyone doesn't think it matters whether you breath thru nose or mouth, here's a little " unscientific" test you can try ;)
Do your normal workout breathing the way you normally would. If you've learned better, your breathing in thru nose, out thru mouth. If not your using a combo of both and as you get fatigued, your probably mouth breathing exclusively.
The next time you do the same workout, get some of those nose clips the swimmers wear, and do your entire workout just breathing thru your mouth exclusively.
If your honest I believe the majority of people would say it's much harder without some nasal breathing, and maybe even impossible with just mouth breathing.

Like MT, I don't claim medical expertise on this subject, but I do have plenty of experience with the subject.

From 35 to 50, I did 5-7 Aerobic/Spinning/Step/ Double-Quad step classes a week. 1 hour, 5 minute warm-up- 45 workout-10 minute cool down and stretch.
When I first started I was a 50-50 person. Started breathing thru nose, as I fatigued I started gasping for air thru my mouth. In the beginning I blamed it on simply being out of shape, but as I progressed and made friends with the different trainers and instructors over the years, they took me under their wings and got me eating right and learning how to breath properly. That put me over the hump so to speak. I could finish a class as strong as I started, something the " mouth" breathers never could ;D