A real answer.
820-2915 2011 is what you have. This is 2011. This board is ok besides the GPU. It will fail every 2 to 3 years, and only get worse as time goes on. The boards available have already been used for 2-3 years, and the replacement chips are harder to find new, so more people will be advertising "repairs" for this board that don't actually work more than 2 months, because they are not using new chips. This board is bad.
820-2850 2010 will fit, but only if you get the older battery. You can just chop up the connector and splice the wires from your battery onto the connector of an older battery, but this is a pain in the @%^. 820-2850 also have rampant kernel panics due to issues with VRAM pgood signals and other that is too much to mention here. Short story before dan calls my sentence noise, 820-2850 is junk to be avoided at all costs. This board is WORSE!
820-3330 2012 will fit but require you mess with the LVDS cable to get it into the connector, which is slightly different. This board is much better. You get a cooler running processor, a cooler running, MUCH MORE durable GPU that will NEVER die on its own, and no stupid random issues. This is also an upgrade in processing power, GPU power, and battery life because this is a more efficient processor. Your present battery will fit this just fine. You may have to replace the LVDS cable for it to fit if you can't jimmy it in there. 99% certain this will have to be replaced.
Any other boards like 820-2330, 820-2523, 820-2532, will not physically fit the machine. Also, these boards do not use the same connectors for wifi, bluetooth, webcam, hard drive, etc - it will all be a mess and not fit at all.
These are the ONLY differences in terms of how it connects and fits into your machine. Make your choice wisely!
P.S.
99% chance Apple is full of $@$* on the logic board issue, I would try booting with a different drive and see if it does the same thing. Apple rarely does real diagnostics in store, just tries to get you to agree to the highest tier of service and politely tell you to F off or buy a new laptop.