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Atmospheric refraction of the light from a star is zero in the zenith
and other sources seem to agree.
I don't see why this should be so. Consider a situation in which air gets warmer from left to right. Then light should curve towards the warm air, and a star at the zenith would appear refracted.
So why do so many sources claim there is no refraction at the zenith? What do they mean by this?
To predict the position of a star one has to account for atmospheric refraction. In a simple model, the atmosphere is treated as being composed of layers of gas with variation in density and temperature due to height, but no variation with latitude or longitude and no turbulence. In such a model, one can predict the position of a star to a great deal of accuracy (so it is a useful model), and in this model there is no refraction at the zenith (as the light doesn't pass at an angle through a medium of varying optical density).
In such a model there would be no twinkling, however modelling twinkling is not needed for positional astronomy. Large variation in optical density that could cause stars to appear significantly displaced at the zenith doesn't occur. The variation in optical density with height is much more significant for positional astronomy.
tl;dr: Atmospheric refraction is different at radio frequencies than at optical frequencies due to water vapor and electrons, and needs to be though of in a different way. It can be much stronger than for optical observations, and it can indeed be non-zero at the zenith! due to ionospheric effects.
The quoted passage in the question should be only applied to optical observations where the refraction is very weak and doesn't vary much (on average) across the aperture of one optical telescope.
Refraction happens to all electromagnetic radiation, not only just visible light.
At visible wavelengths the atmosphere is only very weakly absorbing/attenuating, so the refractive index is treated as a real number, with n ~ 1.0003 due to the dielectric properties of mostly N2 and O2 gas at the surface and dropping with decreasing density going up.
However things are very different at other wavelengths!
For radio astronomy there are (at least) two more sources of refraction:
and the interaction of radio waves with these is more complex.
Water vapor is fairly strongly absorbing in the microwave and infrared wavelengths, with a complicated distribution of vibrational and rotational states that strongly interact with electromagnetic radiation due to the molecule's strong polarization. And whenever you have absorption at one frequency (imaginary part of index of refraction), you have changes in the real index of refraction above and below.
Since the distribution of water vapor with altitude differs from the distribution of the other gases, refraction can vary differently than a model for visible light refraction would predict. Since the refraction can be stronger than for visible light, accuracy become much more important as well.
For very high resolution inteferometric radio telescope arrays like ALMA for example, there are differences in refraction seen by each dish, and so every dish is equipped with its own precipital water vapor monitor which I believe is some kind of broadband microwave monitor sensitive to the columnar density of water in the direction that the dish is pointed, but I'm not sure. So I've asked What is precipitable water vapor in millimeter-wave radioastronomy and how is it measured? and I'm still waiting for an authoritative reply.
The Earth's atmosphere includes a layer called the ionosphere, and for every ion there's a free electron as well. The same way that the electron "plasma" in metals block and reflect light with frequency below the metal's plasma frequency, the ionosphere blocks and reflects radio waves with frequency below the ionosphere's plasma frequency. It's more complicated because the ionosphere is modeled in layers, but you can imagine a frequency somewhere between roughly 10 and 60 MHz depending on the time of day and the mood of the Sun below which the ionosphere is opaque.
At higher frequency and shorter wavelengths than this, the ionosphere will be mostly transparent, but the real part of the index of refraction will still be significant because it is close to an absorption edge.
In this case refraction can be much larger than for optical telescopes. I'm still working on an answer to [How large does refraction become in radioastronomy?](How large does refraction become in radioastronomy?) While @RobJeffries points out that for GHz type frequencies and above the refraction is not much different than for optical, it can get much larger at much lower frequencies near the plasma frequency.
Slide #29 of LOFAR Calibration of the Ionosphere and Other Fun Things shows that refraction can be several degrees at elevations that are still very useful for radio astronomy.
But to address the OP's question directly, have a look at slide #21! A radio observation with an array of antennas has been fit to a simple model for the ionosphere with thin "screen" of refraction at 200 km. It shows a phase shift gradient of roughly 100 degrees over only 0.2 degrees in latitude, which is exactly the kind of refraction shown in the OP's question!
