The term "color" is a label that humans have assigned to denote the ratio between the intensity at various wavelengths in the three different wavelength bands, or regions, that the human eye is able to perceive. These bands are centered roughly at 430, 545, and 570 nm, but are quite broad and even overlap:
Human cone response, normalized to the same height. In reality, the response of the blue cones is significantly smaller, and the green is somewhat larger (from Wikipedia).
If an object emits light only at, say, 450 nm, the ratio is roughly 0.1:0.2:1 (in the order R:G:B); it then looks a special way to us, and we call it "blue", or maybe "violet". If it emits at 550 nm, or 650 nm, we call it "green" or "red". An object that emits light in a more continuous spectrum that covers the region 500–600 nm, we'd name something like orange-/brown-/olive-ish, depending on the exact spectrum.
The Sun emits photons at all wavelengths, but not in an equal amount at all wavelengths. The particular ratios between the three bands that we can see, we have labeled "white". However, when the Sun's light enters our atmosphere, some of the light is absorbed, especially at the blue wavelength. Filtering out the blue results in a spectrum that looks more orange to us. The figure below shows the Sun's "true" spectrum (in yellow), and the spectrum seen from the surface of Earth (in red):
The Sun's spectrum measured outside our atmosphere (yellow) and at sea level (red) (modified image from Wikipedia, with data from Global Warming Art).
Sometimes we want to observe the Sun in a wavelength region that is invisible to humans, for instance in UV or X-rays. This can be done with a telescope and a detector that is sensitive to light in that particular region, but in order for us to see it, we represent the image with a color that we can see. The image in the top of the link you provide is taken with the European spacecraft SOHO's instrument EIT at 19.5 nm, which we call "extreme UV", bordering on soft X-rays. Since this is invisible to humans, they arbitrarily chose to represent it using green. They might as well have chosen pink or brown.
The Sun in Extreme UV, during a particularly violent solar flare (from the SOHO gallery).
Several of the photos in your second link are images taken by the Japanese space telescope Hinode, which observes both in the optical (i.e. visible by humans), X-rays, and far UV. If these are shown in orange, again it's just to make them visible to us, and you may say that they have been "doctored to meet our expectation". In this way, I like better when they choose a color such as all green, so we know it's "false color".