MacBook Pro rapid battery degradation
Most likely, the battery is the issue. However, there is no way a battery should last only 65 cycles before it died. Most likely you have a defective battery. 

Even if you don't have a warranty, Apple will probably give you a new battery if you complain, since the current battery died way before it should have. They usually cover obvious defects even if the computer is out of warranty.

The other possibility is that the previous owner had something done to the battery at some point, and the work was done incorrectly. 65 cycles seems abnormally low for a 2012 laptop, unless the previous owner never used the computer.