2003 S-10 defroster

So, you are saying that you can control the temp when the air is directed to the floor but not when it's directed to the defroster?
Is this an auto temp control system?

You haven't answered my question concerning the symptoms. The site also filtered something out of your answer.

This just doesn't add up to anything logical.
Auto temp control is when you can set a specific temp and everything is automatic from there including the blower speed.
The temp control is done with a blend door but whatever temp you get in one mode should be the same in all modes (floor, defrost, center)

No, it would be normal for the compressor to run in those modes. So, you have confirmed that you problem is that you just have no heat at all.

With the engine at full operating temp, locate the 2 heater hoses where they go into the firewall and feel them to see if they are both hot to the touch. If not too hot to hold, then you have to look at the actual engine temp and if that is up, if it is, then your looking at a restricted heater core.
If you find that they are both too hot to hold, then the heat in that vehicle is controlled by a blend door that regulates heated air flow. It is operated by an electric motor/actuator. The problem can be that the actuator is stripped or inoperative or the door itself could be damaged. This is what needs to be determined by examining the actuator and see if it is responding to heat change commands or not.
If they are both not too hot to hold on to, then you either have a restricted heater core or the system is not completely full and getting air pockets for some reaso