I’m currently a high school junior who wants to major in a math field or physics, and possibly astrophysics. At the moment, I’m looking at Notre Dame as an option for college, but they don’t have an astrophysics major: the have a physics major that lets a student focus on a specific topic like astrophysics. That allows you to take astrophysics classes while pursuing a physics major but not actually majoring in astrophysics. I would like to attend grad school afterwards for astrophysics as well if I decide to go down that path. Would this not-completely-astrophysics major hurt me as I apply to grad school?
It's fine and probably even optimal. for one thing, you don't know for sure that you will want astrophysics four years later. And you are even open to math versus physics.
Do the general physics degree. Gives you most flexibility. You can take a class or two in astro topics, but pick up general physics degree. You don't even know if you won't like some other area when you learn it more (optics, solid state, etc etc.) But in any case general physics program will serve you well for detailed astro study later. Make sure you get a solid grasp on mechanics your junior year, as celestial mechanics is an application of general mechanics
Since you already have some interest in a subfield of physics, I recommend not to major in math. The nineteenth century math (hard calculus and diffyQ computations) is alive and well in physics departments but has been killed, buried, and pissed on, in modern math departments. Also read why Dick Feynman left MIT math major.
No, no one cares. Admissions committees are well aware that different programs offer different things and use different names. The only things that matter are that you have a physics degree with good grades from a reputable school, and research experience (preferably in astrophysics).