Won't Start

1988 Pontiac 6000 2.5. I've been trying to sell this car, and I think I may finally get it sold. (Assuming I can figure out the problem.) Here's the problem. A few weeks ago, we had to charge the dead battery, and we had to remove the battery to charge it due to the fact that the battery uses side posts. Anyway, we had to keep tightening the battery until we got the dome lights. The possible buyer then took it for a test drive.

I have gone over and started it every week. Last week, I drove it for a little while. Now when I started it today, it turned over fine, but whenever I put the heater blower on high, the voltage light dimly came on. I turned the heater off, thinking perhaps the alternator is going bad. I let the car run for 10 minutes, then it suddenly entirely shut off like I turned off the car. All the lights went out as well. I did not see any more voltage warnings before that, and the engine seemed to idle fine until it shut off suddenly.

From that point on, the dashboard lights and the dome light were still bright when turned to the 'on' position (and I could even hear the fuel pump), but as soon as you tried to start the car, it wouldn't do anything at all, not even a click, and the lights all shut off. As soon as you stopped trying to start the car, the lights came back on again, just as bright as before, and I again could hear the fuel pump just like any other time before you start the car, so there still "seems" to be a moderate amount of juice in the battery. Do you think it's probably a loose connection or do you think it might be the alternator? Thanks.