[Galaxy Note II Accessories] rhd-android:
Frankly, my testing is to some degree irrelevant. You should just buy an Anker. Although I haven't tested it yet, I suspect that the Anker battery for the Note 2 matches its stated specs, and ties the OEM / stock battery. I've tested Ankers for two previous phone models (HTC Sensation, and Samsung SII), and in both cases they met their advertised specs. My search for good, cheap, batteries, is largely academic / based on curiosity.

In terms of "better specs" than stock, I'm convinced that this cannot be accomplished unless:

A) You use a physically larger cell.

B) Enough time has passed, and your phone is old enough, that battery technology has advanced.

You don't want to do A, and the Note 2 isn't in category B yet. When I mention category B, the phone that comes to mind is the Infuse 4G. It's a two-year old phone that shipped with a 1750 mAh cell. Well, cell technology has improved in two years, and luckily, this phone seems to also fit some of the cell used in the S2. That means owners of the older Infuse 4G benefit from the advances of cell technology, and luck out by virtue of the fact that cells from newer phones, fit theirs. This won't occur for the Note 2 any time soon.

The reason you're not going to find a higher capacity battery with the same form factor as the stock battery, is simple - if there was a better battery technology available, Samsung would have used it. I recognize that companies don't always use the best technology available, but when it comes to batteries, in the smartphone market, they do.

Battery life is a very hotly compared specification, and every reviewer includes extensive discussion of how much usage a full charge provides them with. Smartphone makers also put immense amount of resources into miniaturizing every aspect of their hardware simply so that they can include larger and larger batteries. That's why they've done things like cut the size of SIM cards down; to create more space for batteries. If there was a way that they could squeeze even an extra 5% of battery life into the same physical geometry, they would do it.

Further, you have to recognize that the battery isn't an expensive component for someone like Samsung. They're a huge manufacturer of batteries in fact. The battery in an iPhone 5 represents 2% of the hardware component cost, or about 1/200th of the retail price of the phone. If a higher capacity battery technology even tripled the cost of the battery in the iPhone 5, that would only be an increase in cost equivalent to about 1% of the retail price tag.

(http://www.zdnet.com/blackberry-z10-teardown-reveals-a-higher-bom-than-apples-iphone-5-7000011558/)

In other words, there is absolutely no reason to think that a better battery than the stock battery is on the market. Energy density is a physical constraint that limits the amount of capacity you can fit inside a given volume. Energy density can improve with better battery technology, but the first people to adopt those better battery technologies will be companies like Samsung. 3rd party manufacturers may be able to tie the stock battery's capacity, by using the same cells, but you wouldn't expect them to be able to beat the stock battery's capacity. In fact, it wouldn't surprise me to learn that companies like Anker source cells from major manufacturers like Samsung, Sony, Panasonic, Sanyo, etc.