Your interpretation of the Carno circle is wrong. At the end, it says that there is a state function entropy. It does not say, that the entropy is a function of temperature only.

What is the reason for switching between the two different definitions of dQ?

It's better not to speak up if you don't talk about the substance.

dQ is not a function of temperature in the Carno cycle.

You take some equations from nowhere, this is your problem.

This is a screenshot of the book, I write it as Q=f (T), and the book writes it as Q=f (θ).

Last time I also encountered someone who needed to explain the content of the book and had difficulty understanding what others were saying.

Q=f (T) is the mathematical expression of "Carnot efficiency is only related to the temperature of the heat source". DQ=df (T) The second law of thermodynamics in mathematics (S) is derived from this and the definition of dQ cannot be changed arbitrarily.

In fact, the second law of thermodynamics changes the mathematical representation of dQ.