The term 'anchor babies' refers to the children of illegal aliens or nonimmigrant kids born in the US while their parents are here studying or on vacation..
It has nothing to do with children born to legal permanent resident aliens. They're already admitted for life and in no need of any anchor.
I'm old enough to remember the term "anchor babies" being used for babies becoming U.S. citizens for being born here regardless of if the parents where here legally or not. See comment # 27 where I go into more detail of an example where folks on the right side of things have tried to pass a law stating that Congress interpreted the 14th Amendment in a way that future Vivek's and Rubio's wouldn't be citizens. They've tried to pass it many times (especially Rep. King), with a few years ago the hope being that Trump would be the prez to sign it.
As a related topic, but similar situation, Boris Johnson (recent PM of UK) was born in NYC, but was never counted as a U.S. citizen. His British parents never pursued that -- they filed the paperwork at birth to confirm his British citizenship. IMHO that's exactly how both Article II's president requirement of natural born citizen should be interpreted, as well as the 14th Amendment's citizenship clause. Just like Boris was a British citizen at birth because his parents were Brit citizens, regardless of where he was born, American citizens at birth are due to having American citizen parents with the location of where the kid is born having exactly zero bearing on citizenship (including presidential requirement).