Hi el9292 -
Welcome to SR. It's a great, supportive community.
I wouldn't worry too much about labels. One question that people use as a guideline is 'do I continue to use alcohol despite the fact that it has a negative impact on my life.' You're the best person to make the determination if alcohol is having a negative impact and that you'd like to make a change.
Alcoholics come in many shapes, sizes, ages, and forms, one common form being not having the 'off' button as you've experienced. It's also a progressive disease, meaning that binging today can turn into daily drinking down the line.
Counseling is a great tool. Perhaps you could use the time with your counselor to unlock the question of whether or not you're an alcoholic?
It sounds like your parents are reacting to one of the scariest things a parent can experience: the loss of a child. Pushing them away probably isn't helping to reduce their concern. Working with the counsellor to understand what's best for you & your wellbeing will help everyone heal. There are many tools available besides AA & inpatient if you decide that you want to change how alcohol is affecting your life.
Lastly, many 'type A' high performing folks have a higher tendency to develop addictions. The same drive that makes us so successful in other ways works against us here.
Good luck! let us know how it goes tomorrow