Battery light stays on most of the time

Make sure the battery is mounted securely and the positive and negative leads are connected securely at both ends.
Positive end should go to starter solenoid and the negative to the engine block from the battery.

Try testing the voltage of the battery with all accesories on and while starting the car.
It may have a weak cell or the anode or cathode may have an internal connection problem.
Has the battery ever seen a hard shock? Like if it was dropped or the car it is in was in a wreck.
By the way, when you load test the battery, checking voltage while starting. The voltage of the battery should stay near 12 Volts. If the Voltage drops drmatically, you have a bad or less than within specification cell. With all the acessories on including the heater blower motor on High, the Voltage should be 13.5-14.1 Volts when the engine is running at 2000 RPM.

When you corrosion on battery posts there is high resistance within the battery.
The high resistance comes from a loose battery cable to post connection or an anode/cathode problem within the battery.
It is an indicator of a problem.
The anode is the positive lead. So, you had some problem on the anode side of the battery. The cathode must be oaky.
Clean the positive post and cable by first removing the negative lead.
Then remove the positive lead.

Get some baking soda and pour it on both post cable, put rag underneath. Get a toothbrush or, if needed, a wire brush and steel wool. Get steel wool from hardware store because pads with soap in them, like Brillo pads, will not work. Get a finer grad of steel wool. Get some rags, battery post cleaner, if you can't find this get brake cleaner,
Let the baking soda sit for 5 minutes, then dip toothbrush in water and baking soda and scrub post and cable. If this does not seem to remove corrosion, use wire but but be a little more gentle.
Get it as clean as you can.
The only 2 surfaces that need to be really clean are where the post and cable mating surfaces touch.
Use the steel wool to shine up these areas until shiny.
Get either ant-corrosion washer or battery anti-corrosion spray.
Use brake cleaner to clean off all of the baking soda.
Check the positive battery post connection to make sure it is tight, because this was the problem post, and stays tight if you move it around a little bit. If it is loose you can get battery post shims that go over post to tighten connection. Sometimes you can spread clamp at bolt tightening area and file it to tighten it up.
When all is clean and checked, put on washers, if you got them, or install the positive lead first. Then attAch the negative lead. If you bought spray, spray it on now.
Keep an eye on the terminals for corrosion. If it does not re-appear, then the battery was the problem.
If it returns, there is a loose connection or corroded connection somewhere on the positive cable.
If you cannot find any of these issues, it is not uncommon for posts to corrode.

I am very glad it worked out.
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