I (24F) am changing my name legally soon and I want to tell my family… how do I have this conversation?

I’ve basically done a major 180 with my name, I’ve decided to change both my first and middle name because my birth name was given to me by someone I am no longer connected with. I’ve been using this name for nearly a year now and everyone but my close family knows and it’s making me feel like a fraud. As such I want to have a name I will be proud and fold of as well as showing appreciation to the parent I love most dearly.

(Context: no longer speaking to my birth giver (50F) and only speaking with my father (61M), haven’t spoken to her in 3 years now and I plan to continue this trend)

My father growing up had 2 nicknames that he’d call me and I love those names, so much so that I’ve taken my first name as a combination of those two names, I love this new name, it’s unique and I don’t think anyone else has a name like it, it’s not a stupid name but a name that makes people go “never heard of that before but it’s pretty cool.”

I’ve been struggling to find the right words over the past year to tell them (my brother as well (22M)) that I don’t feel like my legal name, it just doesn’t feel like me, doesn’t feel right. I haven’t transitioned, I’m still myself but I just feel like this new name is a representation of what I’ve become while away from the grasps of my birth giver.

I don’t want them to see it as a middle finger to them but as a I want to be my best self and the best version of myself and this is how I do it, I’m still always going to be the person I was but just as a different name, it doesn’t change me as a person but simply gets rid of an attachment I don’t want anymore.

So how do I tell the people I love most that I’m not who they think I am?

TLDR: I don’t want to hurt my family’s feelings when I change my name, how do I have the conversation to break the news?