Quantum field theory predicts that light can scatter off light, an effect beyond classical electrodynamics. Maxwells equations are linear such that two photons never interact with each other However, in reality, if you cross two light beams photons will scatter if they're energetic enough. Even in vacuum, the photons... interact with the ever-present quantum fields of charged particles, like electrons, even when no actual charged particles are present But you need very high energies to see this. Thankfully we built a machine specifically to probe nature at very high energies. At the LHC, this rare process of light-by-light scattering has been observed Hidden among millions of photons lighting up the detector, physicists were able to identify 26 distinct light-by-light scattering events This CMS graphic shows how they found them:show more

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That doesn't prove that light is photons. It suggests that light is waves on the aether and that there is a maximum energy/amplitude where the medium aether nomore is linear additive like water has a maximum wave hight above where the waves collapse and break.

Are the photons scattered by virtual electrons/positrons, or by direct interaction with the EM field?

That's what really happens in the double slit experiment. Photons are pushed by photons emitted by slit edges and result of interaction is that particles cannot get to specific locations. Something like this:

Presumably, even a single coherent beam of light has some self-scattering probability, but probably orders of magnitude lower. Right?

Light-by-light scattering happens because space isn’t empty—it’s woven with a geometric and fractal order. This hidden structure naturally guides and strengthens how photons interact. ✨🔄 Instead of moving randomly, photons follow a self-repeating, structured fabric of space that makes their collisions more likely. 🌌📐 At higher energies or near extreme environments like black holes, this fractal geometry will amplify photon interactions. We could observe more frequent and possibly unexpected scattering effects, revealing the underlying structure of the universe. 🚀💥

It is highly doubtful. How were photons isolated from all other particles? For example, neutrinos? I am not even talking about possible quantum gravitons.

It is extremely doubtful due to the impossibility of isolation from, for example, neutrinos. Einstein was wrong and the universe is absolute. Experimental data on the lifetime of short-lived particles prove the inequality of inertial reference frames. Article linked below 👇 1/2

Is there an easy explanation for why the energies must be so high? Why do they interact so weakly? Does their scattering go via the exchange of electrons?

This illustrates that energy can act within and upon itself. It is energy, creating a dynamic feedback mechanism, of energy acting within and upon itself creating identities, that in turn create the by-product we call consciousness of Self. All energy is conscious. E = C -> mc^2