Batteries die multiple ways. One way is internal high resistance. When you do a voltage measure the voltage will look good, but then you go to pull some current from it and it will drop to real low voltage (possibly zero). As mikidymac said, a load test will confirm that. Or even watch the voltage of the battery while using a load on it (ie the starter).
At this point, you have already confirmed the battery can't give power as an energy source because giving it energy from another source fixes the problem. As for it running AFTER you started it...that is because it is running off the engines power generation (the stator).