The two phenomena Feynman referred to are the $q\vec v\times \vec B$ part of the force acting on an electric charge carrier; and the $\nabla\times \vec E =-\partial_t \vec B$ dynamical Maxwell's equation. In the most general situation when both the magnetic field and the shape of the wire is changing, we have to use and add both terms. That's true in each reference frame because for a complicated space-dependent, time-dependent geometry of the wires and fields, there won't be any inertial frame in which one of the phenomena would completely vanish.
I believe that it's more likely (but not certain) Feynman only meant this thing – that there's no way to eliminate or "explain" one of the terms in the general situation.
On the other hand, the fact that both terms have the same origin is a consequence of special relativity and it was indeed one of the motivations that led Einstein to his new picture of spacetime. It seems somewhat plausible to me that Feynman was ignorant about this history.
The full theory of electrodynamics is nicely Lorentz-covariant and it implies both terms. However, these terms aren't not really the same. The Lorentz force comes from the integral $q A_\mu dx^\mu$ over the world lines of charged particles while Maxwell's equations arise from the $-F_{\mu\nu}F^{\mu\nu}$ Maxwell Lagrangian. So Feynman would also be right if he said that the two terms can't be transformed to each other by any symmetry transformation.
Still, one can make physical arguments that do involve such transformations and imply that the two phenomena are inseparable. For example, if we assume the Maxwell equation, it follows from the Lorentz symmetry that $\vec E$ must transform as the remaining 3 components of the antisymmetric tensor whose purely spatial components give $\vec B$. But then it follows that the force acting on a charged particle, $q\vec E$, must also be extended by the remaining term $q\vec v\times \vec B$ for the theory to be Lorentz-invariant. Or one can run the argument backwards. Still, we are dealing with transformations of two different terms in the action that just happen to have a "unified, simply describable" impact on the EMF in wires with magnetic flux.
The simplification and unity only occurs if we assume that the two different kinds of phenomena are in the action to start with but they deal with the same fields which respect the same Lorentz symmetry; and if we study situations that are "understood" or "simplified" on both sides in which some effects, e.g. the magnetic ones, are absent.
Let me say it differently: if the area enclosed by a wire goes to zero, it's an objective thing that is clearly independent of the reference frame. So one shouldn't expect the shrinking of the area is just a matter of inertial systems; it's a frame-independent fact. On the other hand, it's not shocking that the EMF ultimately only depends on one thing, the change of the flux, which has a simple form although it may have different origin.
I would conclude that Feynman was more right than wrong. Lorentz symmetry operates in both phenomena and it's the same one which is a constraint on the most general theory; however, the fact that both possible sources of the changing flux influence the EMF in the same way is a sort of "coincidence", at least if we use the conventional variables to describe the electromagnetic phenomena.
Here is one way to think about it:
When a charged particle travels in a magnetic field, it experiences a force. If the particle is stationary but the field is moving, then in the frame of reference of the field the particle should see the same force.
Now let's take a conductor wound into a coil. In order to increase the magnetic field inside, I could take a dipole magnet and move it close to the coil. As I do so, magnetic field lines cross the conductor, and generate a force on the charge carriers.
It is a convenient trick for figuring out "what goes where" to know that the induced current will flow so as to oppose the magnetic field change that generated it. In the perfect case of a superconductor, this "opposing" is perfect - this is the basis of magnetic levitation. For resistive conductors, the induced current is not quite sufficient to oppose the magnetic field, so some magnetic field is left.
The point is that the flowing of the current is instantaneous - it happens as the magnetic field tries to establish in the coil. So it's not "Apply field in coil. Coil notices, and generates an opposing field. " - instead, it is "Start to apply field in coil. Coil notices and prevents field getting to expected strength".
Not sure if this makes things any clearer...
Best Answer
My position is the same as Richard Feynman's and David Griffiths's and the wikipedia article's. It's simple: The law "change of flux = EMF" is not universally valid. The homopolar generator (that picture you copied) is a lovely counterexample.
(The law does always work for a loop of thin wire, but does not always work in other situations.)
Instead we should use the laws $\nabla\times E = -dB/dt$ and $F=qv\times B$, which are universally valid.