Integral can also be read as "internal parts of the equipment" and that is its intended meaning in Article 725. If we would use your definition, then there would be no application for Article 725.
There is no question that Article 725 applies to the control circuit for a HVAC system. As far as the transformer not making a circuit a Class 2 circuit, that is exactly what does make it a Class 2 circuit. Look at 725.121(A). A listed Class 2 transformer or power supply is all you need to be able to say that the circuit is a Class 2 circuit.
The code only applies to T&M work....it does not apply to contract work.