What's hilarious is that the source to the article you posted is 10 posts above yours... Gaf -> Internet -> Gaf indeed.
I'm glad it's not every achievement and that developers can customize it (because of HDD space). I really hope no developer ends up creating an achievement clip for the last cutscene of a game to spoil it to other people (or have the ability to mark clips as spoilers). However, there have definitely been a couple of achievements this gen that I wish I would have gotten captured. The best ones are always when you do something ridiculous and get an achievement for that.
Gonna be a pretty cool feature though. The fact that there will be some games doing this automatically will mean that there will be tons of videos on XBL for other people to watch since they don't have to do anything, whereas if you want to create your own clip you need to hop out of the game to do that yourself (share UI on PS4, or Upload Studio on X1).
Achievements/trophies are dumb.
Can recording be turned off on the Xbox One or PS4?
Yeah, there will probably be a setting for it on the X1. As for the PS4, I don't think it does anything automatically, so just don't push the share button on the controller and you'll be a-ok.