Welcome!
You don't have to call her anything. I can see how it may seem disloyal but what you do to better yourself is a private thing.
The first step is about admitting powerlessness over alcohol and acknowledging that your life has become unmanangable. Your mother's name is no where in that sentence.
I was stunned too when I realized that I had been affected by the alcoholism in my family. I have said many times that I thought I was the one who made it out normal. By not address what I should have early in my life it only came back to bute me much later. The dysfunction does not go away...it continues to grow as our live's become more complex.
You don't have to go to Al Anon...there are a lot of ways to heal...but for me it worked. It gave me a support system filled with people who had been where I had been and guidelines to live a peaceful life. What have you got to lose by trying what your therapist suggested? You can always stop going at any point.
Hugs,
JT