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I am getting “service battery” warnings and short battery life despite very low number of battery cycles. Resetting SMC and doing a deep battery discharge/conditioning do not seem to have helped.

Sadly, you need a new battery ;-{ Batteries also wear out by age as well, which is the case here. Here’s the needed part MacBook Pro 13" Unibody (Mid 2009-Mid 2012) Battery and here’s the guide to put it in MacBook Pro 13" Unibody Early 2011 Battery Replacement