FINAL UPDATE: I have found the culprit - the Nest thermostat itself. First, I disconnected the t'stat at the control panel and found out that the fan would no longer start upon powering the system. Then I replaced Nest with the old 'dumb' t'stat (from 4 years ago), and everything works fine now. Now I will try to figure out what happened to Nest: failure, software update, etc., but that's another topic.
So I had this problem with my HVAC during winter when out of blue (after working fine for many years) the blower (fan) wouldn't stop, even after the temperature reached the target and the burners shut off. The burners would start and stop per thermostat instructions, it's just that the blower motor wouldn't stop to run. The only way to stop the motor was to power off the entire HVAC system. Mere switching the mode to OFF in thermostat wouldn't stop it.
I kind of half-solved the issue then by disconnecting the G wire from my thermostat (Nest 1st Gen, see picture below). As a result, the heating mode worked fine and controlled the blower as it was supposed to. I couldn't manually control the blower from my thermostat, but it wasn't a problem for me, so I never got to the bottom of this.
Now, it's time to use A/C, and this half-solution backfired. When I now set Nest to cool, the outside fan/compressor starts working, but the blower motor in the basement doesn't start. If I connect the G wire back in Nest, the blower does start immediately after powering HVAC without any command (similar situation as in winter), the cool air blows fine, but again the blower wouldn't stop after the desired low temp is reached. The difference now is that the cooling mode doesn't seem to be able to control the blower correctly (or at all) as the heating mode does.
I have visually inspected the system, cleaned wire ends and tightened the screws.
Any ideas what exactly could be wrong and how to fix it? I can provide more info on the system if necessary. Thank you!
UPDATE 06/06/2019: Got back to this after travel. Since running a new t'stat wire didn't look very practical in my set up, first I decided to inspect the wiring and found a lot of splicing. I added a picture of a big splicing hub that could explain the color mismatch for wires: HVAC's Green wire becomes Yellow on t'stat, Red becomes Black. White is White (although a bit yellowish due to age). But I am confused about where the Blue wire at t'stat comes from. I can see the Blue wire from HVAC going towards the outdoor compressor, but how it ends up in t'stat I don't know. Unfortunately, I am not able to follow the cables to t'stat for the whole length due to basement ceiling.
Still looking for any ideas on why my fan suddenly started blowing without stopping and how to solve this. Thank you!
