(Because of additional processing done by our visual system, what looks "white" to us is very dependent on the color of the ambient light. But the ~5800 K blackbody spectrum could be called the "most neutral" white to our eyes because that’s what evolution has optimized them for.)
But when did humans first perceive the sun as white? We can only reliably say "sometime on or after 12 April 1961", the day Gagarin went into space. Prior to that, as far as we know, all humans saw the sun as some other colour, most often yellow or red, depending on the time of day.
Since Newton's prism, we had pretty good reason for believing it was likely white, reasons that got stronger in the 19th C with improved theories of optics, the discovery of atomic spectra, etc. I'm curious to know when we confirmed for sure its whiteness, and whether this was before October 1957 (Sputnik) or after Gagarin or somewhere in between.
If the sky is covered with clouds, then the light is white, because the blue sky light is mixed again with the direct Sun light.
If the sky is clear and there is an unobstructed view of it, the light reflected by a white object will be the sum of the light coming from the sky to the object and directly from the Sun to the object, so it will be white. For example this is true for white clouds and white snow.
This, i.e. that the Sun outside the atmosphere is perfectly white, has been understood for a long time, since the 19th century.
If you want to know why that diagram has its white point where it is, it's based on testing a bunch of people in laboratories. It's fairly predictable and repeatable, but it might not match your experience outdoors.
My wife and I had an interesting experience a while back where we walked outside our house mid-morning and everything was tinted an intense bizarro green. This was in New England just over two decades ago, and we think it was due to some extensive forest fires that had been burning in Canada and the airborne ash was somehow filtering specific wavelengths of sunlight.
In terrestrial environments, the average color is typically not much different from the true color of the Sun. The white light of the Sun is split by the atmosphere into the bluish light of the sky and the yellowish direct light, then those are reflected and diffused by various bodies until being collected by an eye or a photographic camera.
If the bodies in the environment are random enough, i.e. there are similar amounts of those which absorb or diffuse different parts of the spectrum (e.g. the environment includes brown earth, green plants, blue sky, variously colored flowers and fruits, etc.), then the average color of the light that reaches the photoreceptors will not differ much from the color of the original source of light, i.e. the Sun as seen from outside the atmosphere.
Also, things that are white can be judged by the amount of light they reflect or scatter as a percentage, independent of the source of illumination. For instance, among familiar objects, a common white ceramic bathroom tile comes pretty close to pure white.
Also, a thing can't be "whiter than white" in nature unless it's fluorescent. So it represents a kind of limiting case that people can agree upon.