How do I tell to the mother of my friend that I don't want to live with her daughter anymore?

What kind of boundries is she breaching? Some are worse than others.

doesn;t make sense, why tell her mother? just tell the girl, either you go or she goes simple,

Unless you have a financial agreement with your friends mother you don’t owe her an explanation. Tell your friend she has 30 days to move out then kick her out. In some states you may need to go to housing court depending on the laws. Here over 30 days you are considered a tenant and tenants have certain rights even without a lease.

Why get the mom involved? Also definitely don't hold back on moving out or whatever you need to do

Tell her you are not a lesbian and now have to move out.

Cut ties. I recently had a similar situation I've had to avoid as well

Don't say anything, just move out, leave a letter behind.