Where can I find original MacBook Pro batteries?
Solution 1:
Apple don't sell batteries as spare parts to customers, and the Unibody MBP's battery is not 'user-serviceable'. Hence the funny tri-cornered screw heads. Anything you find on a third-party retailer's site is likely to be 'grey', even if claimed to be Apple-branded.
(Having said that, I have bought several of such batteries for Unibody MBPs and they have worked well with no problems.)
If you take your MBP to an Apple Store, you should be able to get it replaced, but obviously Apple will charge you 'some money' for this.
Solution 2:
There are several posts already that address the questions you ask:
When will my Mac (13inch MacBook Pro Mid 2013) become a un serviceable product in California?
Quality factors relevant for getting a replacement battery (MacBook Pro (13-inch, Mid 2012))
Contrary to popular belief, Apple doesn't make batteries. Apple OEM's (licenses) their battery manufacturing to a supplier that stamps an Apple logo on them. These same manufacturers will sell replacement batteries without the logo.
Like with everything else, quality matters. Battery manufacturers with a good history and a generous warranty have few, if any issues.
With very rare exceptions, Apple doesn't make available parts for vintage hardware (between 5 and less than 7 years). You're 2011 is 8 years so it's highly unlikely you'll get a genuine battery (or new replacement parts for that matter) from Apple.