2003 Mazda Tribute

UPDATE:
The light works when it's cold outside; but as soon as the weather warms up or sits in the sun for a while, the light stops working.
----------

The turn signal lights on the outside work fine, it's just the little green arrow on the dashboard doesn't blink. The clicking sound works though.
Then after driving around a while, it started working again; the green arrow was blinking. I thought maybe the light bulb behind the dashboard died, but now I'm thinking maybe there's a corroded wire or something?

Also, I drove the car around, parked it for an hour, came back and started the car and the battery light came on. Manual says this indicates a charging issue. Drove around a bit and then it went off. Hasn't come on since.
I just tested the battery and alternator last week with a multi-meter to make sure the voltages were fine and they all checked out, battery was around ~12.6v and while running was also in the 14.2-6 range.

Have had the car almost 2 years now and don't know when the battery was last changed from the previous owner. Thinking I might as well change the battery, but finding replacements is giving me different information. Some retailers are suggesting group 96R AGM, while others are 40R. Apparently both work?

 

OK, so the arrow blinks when it's cold outside, but as soon as the the weather warms up or the car sits in the sun for a while, it stops again.
I'm lost.

For curiosity, I checked the fuse box inside the car, noticed some white specs / dust. Blew it out with some air, replaced the old fuses for the that looked like it had some rust on the blades; thought it might be some corrosion or something. Didn't fix the problem.