I wouldn't recommend Cu extension wires
Thermocouples work on the principle of having two dissimilar metals at unknown temperature (the measuring temperature) and at known temperature (the cold junction). By introducing copper into the mix, you are creating two additional junctions (for K-type thermocouples, you would have Cu/Ni-Al and Cu/Ni-Cu). These extra junctions introduce an additional EMF, unless they are held at 0ºC. Temperature controllers having built-in temperature compensation capabilities are not set up to correct for this additional EMF, as you mention. I would amend the following sentence, which refers to figure 1: "Here, compensation cables are not used and thus mV signal at controller end is proportional to (T1 - T2)." The mV signal measured at the controller will be the sum of a voltage that is proportional to (T1 - T2), and an additional voltage, which is proportional to (T2 - T3). The use of extension or compensating cables eliminates this additional voltage by using materials that are similar in the thermoelectric series to the thermocouple material.