If you buy 2 lamps, you've got 2 of all the 7 pieces. Piece 7 (the T-shaped piece) is doubled and you can switch the pattern for the second set. You'll have 15x color A and 13x color B in the first set and 13x color A and 15x color B in the second set. I'm pretty sure this would enable you to create a rectangular lamp.
The proof in the article is a proof by contradiction. Having equal numbers of each colored square is a necessary but not sufficient condition to form a rectangle.
Having the same number of both colours is not sufficient. 14 Z pieces will give you 28 of each color, but you can't make a rectangular lamp (of any size) from only Z pieces.
