I should try these first.
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I drove the car back to where I live. It was about 20km drive at night (mixed conditions of flat road fast driving, corners, hills and slopes) and it went without any issues and no indication of transmission light.
Then few days later, I drove it on a Sunday evening for a leisurely drive about 5km and here comes the transmission light and it's cooling fan.
Later I disconnected the top connector at the transmission like you said, sprayed contact cleaner and wiped both connector and pins with a cotton swab. Then started the car, still the fan and indicator was on as soon as I started on a cold engine. (forgot disconnecting battery that time)
Today I disconnected the battery, reconnected it and started it. No indicator or the fan.
What's left is to drive it around and report the findings.
Meanwhile, I read the Workshop manual and came across these two pages. I understood that if the transmission is experiencing a failure, there would be shifting issues and no display of the mode.
I also read that the temperature sensor is located within the transmission along with the solenoids, so it will not be a simple replacing like a water temp sensor.
Then what does it say on the next page? I simply cannot understand. Could you please explain to me?
Thank you so much