why did she change her birth name?

Yes, that's exactly why. You've found her out, good sleuthing.

More likely she - or her agent - felt that "Margolit Gyllenhaal" was too much of a mouthful for an actress and I bet family amd friends have a;ways called her Maggie.

Or maybe we could all read her trivia, where it states that she didn't even know that her full name was Margolit, but always assumed it was Margaret. She didn't even know it was Margolit until she was 35 and already famous.So, she didn't change or hide anything, but always went by "Maggie".

In an interview with The Observer newspaper in July 2014, Maggie claimed she didn't know her real name until she was 35 years old: "I needed my birth certificate, and when my dad found it, it said that my name was Margolit. It was a bit of a shock. I'd always thought my full name was Margaret - never heard of Margolit. Neither of my parents can remember how it got on the birth certificate.".

Well, that is just incredibly idiotic. Has she never held a driver's license or even a state issued ID, or had to produce her birth certificate for a passport or any of the other multiple reasons most people need their birth certificate? I find that statement very hard to believe. How was she able to obtain a marriage license without a legal ID for which I believe all states require a birth certificate at initial application? Any school she enrolled in would also require her legal name for registration.

And to say neither of her parents can remember how it got on the birth certificate just makes them ALL appear to be flakes.

Well, I am in my thirties and I never had to get hold of my birth certificate for any reason. However, I am from Germany, and of course ID handlings are different here. You only need the birth certificate once, and that is when your parents apply for your very first ID (in Germany there are things like "baby passports" for travelling to foreign countries, or when they/you apply for your first officil ID as a teen). From then on, this is your official ID (it needs to be renewed every 10 years, but that's it) and all other documents (driver's licenses, wedding licenses, etc.) can be obtained by presenting that one ID.
I don't know how the Gyllenhaals did that, but I guess at one point that name "Margalit" got lost/abandoned. They probably applied for everything under the name "Maggie" anyways, also since she was acting under that name from a very early age on and showbiz aliases can also become your "official name" also in documents. What I am trying to say is, it is not impossible that she wouldn't know, because her "official name" was always Maggie (why would a school need to know anything else than an "offical name", I guess "Maggie" is fine for most school officials/headmasters/teachers, if parents state that that is the name their child goes by), while her "legal name" was Margalit until recently.