Light spectrum vs. color sets
Your citation misses the most important parts of the paragraph. In my opinion, this is the important part:
[...] When we speak of white light, we are referring to ROYGBIV [Link to Wikipedia's article on ROYGBIV] - the presence of the entire spectrum of visible light. But combining the range of frequencies in the visible light spectrum is not the only means of producing white light. White light can also be produced by combining only three distinct frequencies of light, provided that they are widely separated on the visible light spectrum. Any three colors (or frequencies) of light that produce white light when combined with the correct intensity are called primary colors of light. [...]
As Wikipedia states in its simple article about primary colors:
Primary colors [...] are sets of colors that can be combined to make a useful range of colors. The primary colors are those which cannot be created by mixing other colors in a given color space.
For subtractive combination of colors, as in mixing of pigments or dyes for printing, the primaries normally used are cyan, magenta, and yellow, though the set of red, yellow, blue is popular among artists.
For additive combination of colors, as in overlapping projected lights or in CRT displays, the primary colors normally used are red, green, and blue.
In your case, we are talking about additive color models.
Additive and subtractive color models
Additive, as in "all primary colors together are white by adding them". Red cannot be mixed by adding blue to green, blue cannot be mixed by adding green to red, and green cannot be mixed by adding blue to red:
Stolen from Wikipedia
Subtractive, as in "all primary colors together are black by subtracting them". Cyan cannot be mixed by subtracting magenta from yellow, yellow cannot be mixed by subtracting magenta from cyan, and magenta cannot be mixed by subtracting cyan from yellow:
Stolen from Wikipedia
Creating your own primary color set
Stolen from the Internet
Therefore, it is a simple matter of making up your own system of colors that fall within the specification of primary colors. Take the above color-wheel and take any equidistant positioned set of 2-3 colors1, preferably from the outer lane - and voilà, you have a set of primary colors.
1 Even 4+ colors could work. There will be a limit where primary colors suddenly will lose their uniqueness, as the one in the middle between two can get mixed by its negative and its positive neighbours. Depending on accuracy, this could be somewhere between 90° to 0,000001°. To help you visualize it, just imagine a straight line between point A and point C and see how close it comes to point B.
Furthermore, you can try Adobe's interactive color wheel to find different combinations.