If we get a coil which is put in a changing magnetic field that increases at a constant rate, then it is known that this increase in magnetic field induces an EMF, now if the circuit is closed then this EMF creates an induced current that flows in the coil. This induced current - like any other current - creates a magnetic field around the coil. Does this magnetic filed due to the induced current interact with the external magnetic field that changes at a constant rate, hence, changing the total rate of change in magnetic flux affecting the coil and hence affecting the induced EMF value? --Or-- Does the induced EMF value depend on the rate of change of the external field and not the rate of change of total flux?
The induced EMF depends upon the rate of change of the total flux, and not just the externally applied flux.
If a current present in a wire, that current induces a magnetic field. If the current is varying the magnetic field induced by the current will vary. That varying magnetic field will in turn induce an EMF. That is the basis for the phenomenon known as self-inductance.
When a coil of wire is used as an inductor in an electric circuit, it's principle of operation is self-inductance. When a voltage is applied across the terminals of the (stand-alone) coil, current begins to flow. However, that current creates a changing magnetic field, which causes an EMF to be created which opposes the originally applied voltage. The result is that the current in the coil/inductor rises over time rather than suddenly.
So, once again, the EMF induced in a coil depends not only on an externally applied changing magnetic field, but upon the total magnetic field, including that created by the changing electric current in the coil.