Induced emf with no relative motion

It is the change in magnetic flux that induces an emf. That change can be due to either motion of a loop or a change in the flux through a stationary loop.

"We know that the induced current is often explained with either the motion of loop with respect to the magnetic field or vice versa."

It is not always possible to explain an induced emf like this. You have given, in your second paragraph, a case where you can't. A transformer embodies just such a phenomenon.

Faraday recognised that there were two sorts of electromagnetic induction; those in which an emf is induced in a moving conductor, and those in which there is a change in flux through a stationary loop. We now attribute the emf in the moving conductor to magnetic Lorentz forces ($mathbf F =q mathbf v times mathbf B$) acting on the charge carriers, which move as the wire moves. In the other case the emf is due to an electric Lorentz force, $mathbf F =q mathbf E$) on the (stationary) charge carriers – if there are charge carriers present in the loop.

One of Maxwell's equations relates this electric field to the local magnetic field, $mathbf B$: $$text{curl} mathbf E = –frac{partial mathbf B}{partial t}.$$ This can be integrated over a loop to give $$ointmathbf E.dmathbf l =-int frac{partial mathbf B}{partial t}.dmathbf A$$ in which the line integral is around the loop and the area integral is evaluated over any surface bounded by the loop.

Some would say that the changing magnetic field gives rise to the electric field, but I prefer to think of the Maxwell equation as telling us an inherent structural characteristic of a single thing: an electromagnetic field, of which $mathbf B$ and $mathbf E$ are two parts.