𝑄𝑐 is exactly equal to the amount of work 𝑊3 we put in during the
isothermal compression, i.e., 𝑄𝑐=𝑊3. Therefore, 𝑊 in eq. (1) can
be made bigger by reducing 𝑊3, in other words 𝑊 depends on us.
Keep in mind that in order to reduce the isothermal compression work (lower $Q_c$) you need to lower the temperature of the low temperature reservoir, $T_c$. That increases the Carnot efficiency ζ in equation (2)
Now, eq. (2) says a completely different story. Given 𝑄ℎ, which we
already fixed, we no longer have any control over 𝑊. It is a constant
that is determined by the temperatures of the heat baths and the
initial amount of heat exctracted 𝑄ℎ. Where does this apparent contradiction come from?
There is no contradiction. Equations (1) and (2) are not independent of one another. As i indicated above, in order to lower $Q_c$ in equation (1) you need to lower $T_c$ in equation (2). Lowering $T_c$ while keeping $T_h$ fixed in equation (2) increases the efficiency and therefore increases the net work done.
"in order to reduce the isothermal compression work (lower ) you need
to lower the temperature of the low temperature reservoir, " Would you
mind explaining why this is the case? I don't really see why the
amount of work that we put in depends on the temperature of the
To help visualize why this is the case, see the Pv and Ts diagrams for the Carnot Cycle.
The area enclosed in the cycle represents the net work done in the cycle. Note that in order to increase this area, you need to raise the isothermal expansion path and/or lower the isothermal compression path. To do this you need to increase the work done in the isothermal expansion (path 1-2), which means increasing $T_h$ and/or decrease the work done in the isothermal compression (path 4-1), which means lowering $T_c$. Either approach increases the Carnot efficiency and thus the net work done.
Hope this helps.