Is she my friend?

So let me start off by saying I don't want to be the center of her world. I get people are busy or whatever.

I have a friend that I've been friends with for years. She's shy but not so much with me. I'm always inviting her to hangout and do different things. I'm always making plans with other friends of mine and inviting her and introducing her to them. I'm always starting conversations first with her.

Now, she goes out and does things as well but never invites me. She'll tell me after about these events and then say, "I was going to ask if you wanted to come but I thought you would be busy." She'll never start a conversation. If I don't message her then I won't hear from her.

My problem is she wants to be included when I have plans but she doesn't really think about including me. It makes this friendship feel one sided. Honestly now I just don't care if we're friends or not. Am I wrong for thinking this way? Please tell because I also feel guilty for thinking like this. Am I a bad person for this?

Is she even my friend?