I'm not much of a label person, myself. IMHO, folks get way too hung up on whether someone is or is not "really an alcoholic". The question is not whether or not a label applies, it's whether the behavior of the person in question is acceptable. If it is not, then some action is necessary on the partner's part. Maybe that action is to leave the relationship; maybe it's to stay and try to deal with the addiction using CRAFT techniques; maybe it's to go to Al-Anon; but endless musing about the label just doesn't seem to be helpful.
Now, I cannot for a minute tell you what to do. No one really can. I've recovered from my own addiction, so I know that recovery is absolutely possible and that people who were once very much addicted can live good, healthy lives and have healthy relationships.
At the same time, if I'd been in a relationship with ME when I was still addicted, I would have left. I was a mess and it took me years to get to the point where I was ready to quit. I took a lot of good folks with me on my little ride to hell, and they didn't deserve what I did to them.
And it may seem odd, but having gone through all that horror myself, I would not for one minute have a romantic relationship with a person who behaved as I once did, or perhaps to be more specific, with a person who behaved as your wife is behaving. Life is too short for that. When I was dating before my current marriage, I would not even date a person "in recovery" unless that person had been abstinent for at least 5 years. I broke up with someone after he got drunk ONCE. Kind of a no-tolerance thing.
I don't know if this helps, but I certainly hope so.
One thing is for sure, we all deserve to live the best life we can.