Actually, that is exactly what I'm saying.
You are quoting the element of the year branch from the Tzu Ping Ba Zi distribution of elements. Marie is using the Na Yin element of the year. On top of that, we have the Xuan Kong element of the year, the gua element of the year, the hexagram elements of the year and the element of the year branch. That's not counting the hidden elements of the branch, which in the example you mentioned, the rooster, there happen to be none. And just to make life a little more difficult, the cut-off point for the year is different depending on whether you are using the solar or lunar Chinese New Year.
In short, depending on your point of view you could practically be any element at all. And you may appreciate why I gave the short answer confirming that there is no printing error in the system - I know because I checked after I had this very discussion with Marie years ago. To understand how all these elements came about would involve a very long discussion on the rudiments of the Ba Zhai, San He, San Yuan and Xuan Kong Da Gua major schools of Feng Shui, together with the Zi Wei Dou Shu and Ba Zi methods of astrology, as well as a smattering of Plum Blossom divination and possibly Qi Men Dun Jia's mysterious door methods. And even then you will eventually come to, ultimately, "because so-and-so said so".
In the end, it depends on how you apply that knowledge. The application is within the context of the system that uses that calculation. Solar-based Feng Shui masters use the solar elements and lunar-based ones use the lunar, according the application methods they have been taught. As for which system is correct, the schools have been arguing and defending themselves pretty much the way you describe for generations. (That is, before Feng Shui as a practice itself was practically stamped out in China.) I find my time is better occupied elsewhere. I know the methods within those frameworks that work for me and the ones that don't, but that is the result of experimentation, which is probably the best way to do it.