You are confusing additive and subtractive colour mixing. If you mix paints together you should get black, not white.
In additive mixing (as used in TVs and monitors), you create light, which is then mixed. When you mix the three primary colours (red, green and blue), you produce white. Other mixes produce other colours, for example red and green combine to produce yellow.
When you use paints, you are using an external light source (the sun or a light bulb) and each paint reflects some of the wavelengths and absorbs others. For example, yellow paint absorbs the blue wavelengths, leaving red and green, which mix to yellow. This is called subtractive mixing, and the primaries are cyan, magenta and yellow; when you mix paints of these colours, the result is black. Adding additional colours to this mix keeps the result black, as there is no more light to reflect. Other colours are made up by mixing the primaries.
With both additive and subtractive mixing, the result of mixing colours depends on the purity of the primaries. No paints are "perfect" cyan, magenta or yellow, and as a result the mix will not be completely black. You may get a dark brown or purple, depending on the paints you use. This is one (of several) reasons why printers use black as well as CMY.
The same goes for monitors: you never get "pure white" - which is typically defined as light with a colour temperature of 5500K, about the same as sunlight. Some monitors can be set for different temperatures. Some are set to 9000K, giving white a bluish cast. Interestingly, the colours that can be displayed on a monitor do not match those of a printer (or paint). A monitor can display colours that a printer cannot print, and vice versa. Every device has its own colour gamut, usually smaller than the eye's gamut, so with any device there are colours we can see but which the device cannot produce.
The reason why all this mixing occurs is because our retina has sensors for red, green and blue, and the brain mixes these inputs to tell us what colour we are seeing. This is why the primaries are RGB, or CMY.
One half of the problem is light. Light is an oscillating electromagnetic field. The frequency of oscillation determines the color. Higher frequencies have shorter wavelengths and are blue. Lower frequencies have longer wavelengths are red.
Light is a mix of a range of wavelengths. Sunlight contains a wide range of wavelengths, including some that are too long or short for us to see. A laser contains a very narrow range.
How light interacts with matter is the other half of the problem. Electromagnetic fields exert forces on electrons. Electrons in atoms or molecules are constrained in how they can move.
Metals have electrons that can move freely. This makes them good conductors. Freely moving electrons reflect light. Light makes electrons vibrate and absorb the light. Vibrating free electrons emit light and stop vibrating. Thus metals are shiny.
But even in metals, electrons are constrained. For example in copper, electrons can vibrate more readily at low frequencies. Copper reflects red light better than blue. It reflects better still in the infrared and even worse in the ultraviolet. Here is a graph from a random website, http://www.minoanatlantis.com/Minoan_Mirror_Web.php
So if you shine a red laser on copper, it reflects almost all of it. If you shine a green laser on copper, it would reflect less.
Other materials have different ways of interacting with light. But at each wavelength, they may reflect some and absorb some. They reflect more at some wavelengths and less at others. Even black materials reflect some light.
Best Answer
The explanation you give is correct. A white body reflects all wavelengths. We call it white when all colors (all wavelengths) are reflected from an object and hit our eye. Black is the opposite.
I would say that white is all colors, as you do. But maybe he sees it from the perspective that since all is reflected and nothing is absorbed, there is "no light" left. I mean, it depends on what he means by "no light". It could be a matter of definition of the words, so maybe you actually agree on what happens but call it differently.
Nevertheless, your explanation is correct.