I agree with your assessment regarding the intention of the phrase. We're back at the silly arithmetic meme that hinges on not grouping terms explicitly and watching people yell at each other in the mistaken belief that there's one authoritative interpretation of an ambiguous string of symbols.

Still, the actual mistake remains. Why an extra 1/6 of the pizza? 1/3 of 1/3 is 1/9, not 1/6. That's 1/2 of 1/3.