They aren't professions, bud. They're pass times.
I can't be a professional woodcutter when I'm busy being the Dovahkiin, nor can I accept a job at a lumber mill. I'm told that if I'm looking for work, I can chop some wood. I am not told "Hey, we need a hand at the mill. Care to be at work at 8am ever day?"
I'm sorry, is there actually a game where you are hired at a company, asked to come in between 8 and 5 and work at a specific job?
Oh right, that's real life.
In-game professions are usually defined as some kind of job you can do that isn't combat-related, including trading, smithing, enchanting, mining, wood-chopping, alchemizing, etc. That's how it's defined in every other RPG ever, and I'm not sure why the Elder Scrolls should be held to a different standard.