So, wouldn't it be better to write books on how someone got from a problem to a solution and why the outcome was something that can be described as a factory pattern, rather then the other way around?

I didn't read books on design patterns, but the book from the link structures around what design patterns there are and how they got applied to a problem. It'd be nicer to first have a problem and show why and how a particular pattern is better than, say, a bunch of monolithic lines of code.. So that you can say "oh and by the way, the way we solved that problem falls into the category of factory patterns".