Li-ion batteries have a very flat voltage discharge curve. Their voltage does not drop as much as NiMH or NiCd (or even alkaline) batteries as charge level decreases. So it's relatively easy for battery protection circuits to mistake the voltage depression due to the sudden application of load for a battery with no charge.
What happens when you power up on AC, then unplug AC forcing it to run on battery? If it stays powered on, my first guess would be a HDD or fan whose bearing is going and thus draws an unusually large amount of power when spinning up. If it shuts off, my guess would be a fault in the battery charge level detection (calculation) circuit, although it too could be caused by something drawing excessive power.
Was the second battery you tried newly manufactured (check the manufacture date)? Or had it been in storage for several years (either at your house or at the store warehouse)? Excessively long storage times will kill a Li-ion battery due to self discharge. A Li-ion battery which has been overdischarge is prone to catch fire or explode if recharged. So the protection circuitry built into the battery will kill it if it ever senses an overdischarge state, preventing it from being charged or used again. The symptoms you're seeing may be how a killed battery behaves in this particular laptop. If you're storing Li-ion batteries (or NiMH), it's recommended to pop them back into the device every 6-12 months to check their charge state and recharge if necessary. (1-3 months for NiMH, and charge those only to about 50%-70% before storage.)