Is this the starter or the battery?

I started my car this morning, and it cranked very slowly before finally starting. I used the car throughout the day, and sometimes it would start slowly, sometimes start right up.  Then tonight after driving for about 40 minutes, I shut it off to go into the store. It was off for maybe 10-15 minutes, when I got back in and turned the key, there was nothing. No click, no chugging, no sound at all, but all the lights came on fine. Dashboard lights all on, radio worked, interior lights turned on without an issue and weren't dimmed.

After trying a couple of times, I was ready to call AAA and get it towed home, but decided to try one more time. This time it started right up. I left it running after this for about 15 minutes to see if it would shut off, and it didn't, as it also didn't throughout the day, so that makes me pretty sure it's not the alternator.

I don't know a whole lot about cars, but my thought process is: It's not the alternator because it would have died while I was driving, and wouldn't have been able to stay idling for 15 minutes. It's not the battery because all the lights and the radio were on full, and once your battery's dead, it's dead. It wouldn't not start for two or three turns of the key, then suddenly start. So I'm thinking it's a starter issue.

Does this sound right?