First of all, when charging, you really should have a charge quite that high going in. At 14.6 volts you can make it hot, and screw up the chemicals inside. If you have ever watched what your voltage it will run mid 13 up to 14. Shouldn't be higher. If you see your alternator charging like that, you are going to fry your battery and need to check the regualtor.

If you had a reading of 10.4 volts, one of the cells is likely dead. You might have a plate cracking. Get a new battery. It won't hold a charge for any length of time, and it sounded like you charged enough to get the thing going.

second thing.

The relay that controls the shift control lock is a solid state relay that if I recall is near the shift lever on the driver's side, under the felt. It's black.

However, the car will start if the relay failed. It just won't get out of park. I disarmed the system when my relay went bad. It made me mad because it misbehaved on my mother, and my mother is disabled. It wont do it again.