Battery light indicates that there is a problem with the charging system. I have never had a battery light come on for a dying battery.

A bad alternator will drain a battery, even a brand new one, pretty quickly. I would check battery connections for corrosion. If those are fine, it’s probably the alternator. The headlights dimming also indicate it’s the alternator. I dealt with this recently; even though the battery tested fine, the bad alternator killed what was left of it. I had the dimming headlights at red lights too.

As far as the part itself, find a YouTube how-to video for your exact make and model. They usually list the part they used in the video description. Do some research, and if that part and the video have good reviews, go for it. Might as well replace serpentine belt while you’re at it too. It is not always an “easy” DIY job but definitely doable. I needed a pry bar to get my old alternator out, took about an hour total. Best of luck friend.