When I was your age, I had exactly the same problem. It boils down to this:

Your mom doesn't want to contribute to the likelihood (in her eyes) that you're going to get killed on your bike. She'll never be able to forgive herself and will fret constantly that you're lying on the road in a ditch somewhere bleeding to death and it's HER fault;

She doesn't like the idea of anything with tires that have absorbed dirt and grit and grime on the road having any contact whatsoever with her clean floors/carpet;

She's not a bike rider, never was and doesn't know anything but department store bike purchases.

and it's in that order. The first problem is one you will not be able to solve and it is the weightiest of them all so you cannot expect her to help you financially AT ALL. You are angry, justifiable or not you have to accept this.

No.s 2 and 3 can be solved but you will have to let your mom know of your intentions that you are going to have a bike and you will respect her concerns by taking care not to store the bike in such a way that it will damage her precious floors/carpets/walls, etc. Show her how you plan to do this. My mom went so far as to tell me that even if I purchased a bike out of my own money, I'd never be allowed to store it indoors. That was a problem since we lived in an apartment.

After having done all that, you will probably wind up in therapy over control issues between you and your parent(s). You will wonder when you get older and wiser why you let something like this make you angry and stressed out in the first place. It's simply unAmerican to grow up any other way