Faraday's law. And Faraday's law of induction

(a) As far as I know, Faraday's law in electromagnetism is another name for Faraday's law of induction.

(b) You refer to the equation $$\vec{\text{curl}}\ \vec{E}=-\frac {d \vec{B}}{dt}.$$ Applying Stokes's theorem (and taking the differentiation outside the integral sign) this integrates to $$\oint \vec{E}.d\vec{s}=-\frac{d}{dt} \int_S \vec{B}.\vec {dS}$$ The left hand side is the line integral of the electric field, that is the induced emf, in a closed loop enclosing an area S, and the right hand side is the rate of change of magnetic flux through S, so we have $$\text{emf in loop = –rate of change of flux through loop.}$$ So the difference implied in your second paragraph between the curl equation and "emf = – rate of change of flux" isn't a difference at all!

(d) All this is valid whether or not there is a conductor.

The betatron is a particle accelerator that can be understood in terms of an induced emf in a non-conducting toroidal chamber. You could argue, I suppose, that the presence of electrons being accelerated in the chamber makes it a conductor! But it would be weird, wouldn't it, for the emf suddenly to appear when electrons are injected into the chamber?

(e) The equation "emf = – rate of change of flux" can, though, also be applied to a moving conducting loop cutting flux, though this time the emf arises from magnetic Lorentz forces driving charge carriers around the loop.