Tl;dr: Car is having trouble starting, and I'm wondering how to determine if it's the alternator or the starter. I don't experience the longer turn over times of a typical starter malfunction, but I also don't see the dimming lights or no engine rpm of a bad alternator, even after a week's use.
Background info My car had trouble starting ~8 days ago our timeline starts there
Mon, day 1: The engine tried to turn over once then refused to try again. First time car has ever had trouble starting. Headlights still worked, so did radio and other electrical doodads. Seemed to start a little slower after work, but in the coming days...
T-F, days 2-5: No difficulty starting or driving. My work is ~30 minutes away and I take the interstate.
Sat, day 6: I had trouble starting it late at night and needed to jump it. Same style of thing as on day 1. It had been sitting since Friday afternoon.
Sun, day 7: Car sat for the whole day
Mon, day 8: Trouble starting same as before. Advance ran their electrical diagnostic and did not think it was an alternator or battery problem, though they noted the battery needed charging even though I drove ~30 minutes to get there. Advance seemed to think it was the starter because of low cranking amps, even though the battery drains while the car is off. I had them fully charge the battery. I also ran the car for a bit at night.
Tues, day 9: No difficulty starting, but now check engine light is on.
I was hesitant to think it was a starter problem, since I've never experienced longer turn over times, but also doubt an alternator issue because I was able to run the car for so long after the first jump, and it seems to start well if I just drive it regularly.
Is the situation I've described more indicative of a bad alternator or bad starter?