You've started the troubleshooting already. Next check all your power connections really well. Try to wiggle or move the connections. If they move at all you have a bad connection and that can contribute to non-starting. Corroded terminals either need to be well cleaned or replaced.
If you have a meter, have someone turn the key for you while you measure the voltage across the battery. If it drops below 10 volts while trying to start, that's a good indication of a truly dead battery. If you have a dc amp clamp you could also see how much current is be pulled on start. I don't know the specs for a Saturn, but you may be able to look that up online. You may be able to recharge the battery either by jumping it with another vehicle and then just letting it sit there while the other vehicle runs for ten or fifteen minutes. Of course you could just try a battery charger or another battery too. Since the motor does turn, that eliminates your actual starting electronics and the starter itself from being the culprits(usually). I have actually seen a starter with a short in it draw down the battery and allow the motor to crank just a bit, but you said that you replaced the starter. Good luck with the fix.
JJ
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