As a math grad student, I can say that this Chicago list has primarily books that mathematicians know. The one posted here has primarily books that I am unfamiliar with. If one were to follow that one, other people trained in math would have a hard to judging what you've done (it's common to say things like "I've learned algebra at the level of Dummit and Foote" but this only works if people know the book you're referring to).

On the other hand, books written for mainstream math majors and graduate students are not necessarily the ones best suited for an autodidact. Perhaps the author of this post has selected those that are more appropriate, but I can't judge. Also, Springer is a great name in math books and you generally don't go too far wrong by sticking with them, but I've never seen their undergraduate series before. Perhaps they're more common in the UK than the US?