Are you thinking the battery has a full charge because the charger light is green? That light is also green if the battery is missing. Have you looked at the battery sensors on the side of the case? Do they light up?
It sounds like the logic board isn't drawing power from the battery, but only from AC. Your battery may be as much as four years old; if you use it a lot without AC, you may be near (or at) the end of its life. If that's the case, replacement batteries are readily available. There's a special Y-wing screwdriver used to fix the batteries down in Unibody MBPs; iFixit has those as well, either as separate screwdrivers or as swappable bits in the driver kits. iFixit is currently out of stock on this battery model, but both Apple-branded and third-party batteries are readily available; look for Apple Part Number 661-5557.
But let's try a stopgap measure first...
In many cases, if a laptop sees the battery but the battery is completely drained, the computer won't start up until there's at least a minimal charge. I'd try the SMC reset again, and then leave the MBP on the charger overnight without powering it up. That way, the battery may be able to take enough of a charge to be able to start up.
If that works, you can reattach the charger to complete the charge cycle, and open System Profiler to check the battery's condition. In System Profiler>Power, the sections you want to check are:
- Charge Information (what is the full charge capacity, and where is it now? The rating for a new battery is 5600-5800 mAh)
- Health Information (what's the cycle count? A lithium ion laptop battery is usually good for about 1000 cycles, but the battery loses charging capacity as it ages. I'm using a 2011 MBP 13" with 978 cycles; its charge capacity has been reduced to 5157 mAh)
MacBook Pro 13" Unibody Mid 2012 Battery Replacement
[linked product missing or disabled: IF163-019]
Tri-point Y1 Screwdriver
[linked product missing or disabled: IF145-022]
If it turns out that the battery is charging normally, but you still can't power the computer off the battery, nor start with the battery plugged into the logic board, then that's likely to be a logic board issue. Since that's an expensive problem to solve, I wouldn't assume that until I'd tried a replacement battery first.
If you're near an Apple Store, the Genius Bar should be able to check for a power issue without forcing you to commit to the repair. At least if their check reveals a logic board problem, you can decide what to do next.
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