The story goes on to say that
the gene for our red and green cone types lies on the X chromosome. Since women have two X chromosomes, they could potentially carry two different versions of the gene, each encoding for a cone that is sensitive to slightly different parts of the spectrum. In addition to the other two, unaffected cones, they would therefore have four in total making them a tetrachromat. For these reasons, its thought to be a condition exclusive to women, though researchers cant totally rule out the possibility that men may somehow inherit it too.
This is about right, except for the last sentence. (Having only one X chromosome, men could not possibly be tetrachromats.)
The article cited in the OP fails to mention that in each cell in a woman, only one of the X chromosomes is active. Each X chromosome normally has a gene controlling the pigment in red cones and another gene controlling the pigment in green cones. Both types of gene are quite variable in the human population. This makes for some variety in the spectra of red and green pigments. (Technically, they are called long- and medium-wavelength pigments.) A woman's retina is actually a mosaic - like the skin of a calico cat. In some patches, one of her X chromosomes is active. In other patches, the other X chromosome is active. In principle a woman could be a pentachromat, i.e, she could have five different types of cones: two red, two green, and one blue (short wavelength). (The gene controlling the pigment in blue cones does not lie on the X chromosome, so it is impossible for anyone to have more than one type of blue cone.) Pentachromacy probably exists, but it has not been observed. Its effects would be very subtle indeed.