1993 Honda Accord Battery Problem?

1993 Honda Accord 4 cyl 100000 miles

Two days ago my battery light and ABS lights started flickering on and off, but the car ran fine. Yesterday it started in the morning but wouldn't start after work. Lights went on, but it woudn't start. I got a jump and it started right up. There is a lot of corrossion on the battery terminals. While it drove, I had no problems, but the battery and ABS lights were still on. I poured baking soda and water over the battery terminals and most of the corrossion came off. This morning it wouldn't start again. The lights go on, but it won't start. I'm assuming I need a new battery but when I was looking online some people said the battery light actually means your alternator is bad?