1987 Honda Accord Car is trying to crank, but not starting

I've asked a question previously regarding this same car. The problem was the same, but different symptoms. Previously my car wouldn't start, wouldn't even make a sound when I tried to start it. I've had my battery checked (about 2months ago) and have also, since my first question to you, disconnected the battery cables and cleaned up and corrosion. When it was not doing anything when I tried starting it, my dad had made sure it was completely in park and then tried to start it and it started right up. In the last month, when it wouldn't start (only happened once or twice) I'd double check that it was in park and bang it started.
What is currently going on is that it's trying to crank over, but isn't starting (the headlights do dim when I try to crank it over). My question today is, how do I know if it's my starter vs. My alternator or possibly just a bad battery? The headlights stay bright until I try to start it and thing's like the radio, heat, electric windows, etc. Do work.
Thank you very much for for your help, I can't express how much it's appreciated!
~Lisl