We do have to acknowledge the obvious difference in that daylight on Earth includes blue-sky diffuse illumination. We count on this to fill in light in shaded and shadowed areas. And if we're paying careful attention to our photography, we know that this light is hue-shifted toward blue and preferentially filled with UVA wavelengths, so we take precautions.
Because we're used to this on Earth, we tend to think that all indirect light is the result of this sky fill. That's actually far from true. Reflected light from surfaces is also responsible for quite a lot of shadow and shade fill. You can demonstrate this to lay people using studio lights, etc. But experienced photographers have already figured it out. They know how to fill in shadows using white or silver panels to reflect light diffusely into places where it's needed. My set designs for theater have to account for lighting reflecting off the stage floor or set walls.
Although the front of the lunar module was facing directly away from the sun, you can see the shaded side in photos because it's 7 meters tall and thus has a view factor to the sunlit lunar surface to the north and south of the LM's shadow. The lunar surface preferentially retroflects, so you get even more light.
Shadow is different. To be accurate, shade is what happens when a surface is facing away from light sources. No light from the light source reaches it directly, so it's shaded. Shadow is what happens when something is between a surface and a light source. Sometimes the difference matters and sometimes it doesn't. But both require the viewer to understand the concept of view factors.
Shadows cast in sunlight onto the lunar surface often don't have a view factor to any other lit surface. This is because the shadowed surface is flat ground with a view factor only to the black sky, the sun being obscured by the object casting the shadow. Sometimes the shaded side of the shadow-casting object will be indirectly lit, and this indirect light will fall further indirectly into the shadow. But this wasn't enough, for example in the LM, to allow detail in the LM's shadow to be easily photographed. The earthshine photo just posted above illustrates this—not only that, but earthlight is optically significant on the lunar surface. Earth was waning gibbous during the Apollo missions.
But when you go to photograph a shadow, it's possible for you to be the fill. If you're standing over a limited shadowed area, and you yourself are sunlit, you can provide shadow fill. This didn't happen too often in Apollo photography; most shadows are, in fact, pitch black. This is because the side of you that's facing the shadow is usually your shaded side. If I shoot directly down-sun, the shaded side of my suit is facing the shadow. Any fill into the shadow would need to be a double diffuse reflection: surface to suit and back to surface.
Here's Aldrin carrying some equipment.
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