(I'm not sure why but I wrote this in 2022. I found it and want it preserved here. Also, some of it is not quite right. Enjoy.)
You are correct in the idea that the photons that are reflected and not absorbed is what causes us to see color. The idea "what it passes through gives it its color" is not right. The color of a photon is determined by the amount of energy that goes into its creation and it never changes.
Absorption and reflection aren't exactly what they seem to us, though. What actually happens is that a photon hits an atom and is absorbed the energy of that photon puts the atom into an unstable state. This causes the atom to spit out a new photon to get back in balance.
Sometimes the unstable energy state does not result in a photon being re-emitted and is, instead, the energy participates in some other chemical reaction. Sometimes, there is energy left over from that reaction.
That energy is re-emitted. The new photon will have that amount of energy, ie, it will be a new color. The original did not change. It contributed to the creation of a new one.
This is what happens with photosynthesis.
Roughly speaking, chlorophyl molecules in plants absorb the light that hits them. That puts each molecule into an unstable energy state but, instead of re-emitting (reflecting) all of the energy, it keeps part of it by changing in ways that are part of the photosynthesis process. The part that is not used has the amount of energy to re-emit a green photon.
You mention white light as "pure". This is not really correct.
White light is not actually a color of light and it's sort of the opposite of pure. It is always a combination of photons with different energy levels contributing to a sort of illusion created by the chemistry of our eyes.
Our retinas have three chemicals that absorb photons. That means the the energy they absorb with the photon participates in a chemical reaction. That reaction causes nerve signals to our brains that we interpret as color.
Each of the three chemicals responds to photons in a narrow range of energies (which are also, btw, equivalent to wavelength). We call the experience we perceive from those chemicals, red, green and blue.
When all three chemicals absorb roughly the same amount of photons, ie, equal amounts of reg, green and blue, our brain interprets the signal as white light. The photons are not white, there is no such energy level. It is a bunch of different photons that our brains interpret as 'white'.
We cannot see infrared because the chemicals in our eyes do not absorb infrared light in a way that creates a nerve signal. Instead, those photons cause the molecules to vibrate in a way that we interpret as heat. That's why warming lamps in restaurants shine infrared on the food.
Sorry to go on so long but I find this all so fascinating.