Saying that all loop antennas work because of Faraday's law is like saying all electric circuits work because of Ohm's law. It doesn't mean much.
Small loops have maximum sensitivity or radiation in the plane of the loop (narrowside). The sides that are parallel to the incident electric field will contribute to the electromotive force and the other parts (perpendicular to the incident E field) are passive conductors. However, the size of the loop is much smaller than the wavelength, so the current on the entire loop is rather uniform. The loop functions as an antenna because of the different phases at which the target signal arrives at two parts of the loop.
Of course, the parallel explanation in the magnetic field is that the incident magnetic field can be viewed as a magnetic dipole moment in the loop.
But both of these things happen simultaneously when the radio signal (far field) is concerned.
Turning to a full wavelength loop. The sensitivity maximum is perpendicular to the plane of the loop (broadside). The fed and its opposite sides have the current induced by the fields, and the other sides have passive currents. There is no phase difference in the fields incident on the two "active" sides. The "delay" needed to add up two elements' antenna electromotive forces is created by the passive sides. There are also standing waves around the perimeter because it has a non-uniform current distribution, unlike a small loop.
As a side note, folded dipoles, mentioned in the other answer, is best described by the transmission line nature of the folded elements; two conductors have equal but opposite currents. That is because of the conductors' proximity and intersecting the same field together. Even and odd mode currents on the transmission line define the antenna behavior. As such, different models explain full wave loops and folded dipoles, and I disagree with the other answer.