I'm running a Verizon Note 2 so I can't speak from experience, but I wouldn't really buy into the battery drain thing. Here's why:
People talk about battery life being better on one ROM and worse on another. Over the years, one thing I've learned is that any changes from kernel/ROM base etc that are not major ground breaking bugs, will have minimal impacts on battery life. So insignificant are these changes, that what people perceive to be battery life changes are nothing more than placebo. The only way to be sure there's a battery drain problem is to use logged data that shows objectively how battery life is worse than it was before. Even then, there are so many variables in play to define just how long your battery life will be from charge to charge that it's almost impossible to properly gauge battery life from one ROM to another.
That being said, I have yet to see any hard evidence showing severe battery drain issues. If there was one, you'd know. We're talking an hour or two less screen on time per charge, not "well I had to plug it in by this time, as opposed to by this time, so I am seeing worse battery!!!1one!1" Not likely.
But your post more importantly, for me anyway, shows something that I like to see posted. A user experiencing overall a better quality product. You had your phone, it was all good and enjoyable on 4.1.1, then they sent you a free OTA update that BETTERED your experience with that phone. For some odd reason, there are a TON of people, particularly on the Verizon board, who feel that they never want an update for their phone at all. They fear 4.1.2/4.2.2/etc as if it's some kind of virus coming to infect their perfectly operating device. The logic blows my mind. Serious bugs aside, which arguably don't occur often at all, there is only good that can come from a new update.
Glad to hear you're enjoying 4.1.2, and I wouldn't pay any mind to the whinges from people going on about battery life. If there was a serious problem, you'd know. And I don't believe that's the case with 4.1.2.