I'm wondering what is the term for the person, that is doing business on his/her own and has tax liability. It's different in each country, I'd like to know how they are called in USA and UK.
Sole Proprietor: the person who owns and controls a sole proprietorship.
I am the sole proprietor of this business.
Independent Contractor: A person working independently, under a contract; a self-employed person.
I am an independent contractor.
Self-Employed (or Freelancer): working for yourself.
I am self-employed.
I am a freelancer.
The question is vague.
In the U.S, "sole proprietor" is a legal term that signifies the one human being who is responsible for the debts of a business and legally entitled to the profits of the business. A sole proprietorship is distinguished legally from partnerships and corporations, in which the debts of a business are not the responsibility of a single, "natural person," another term of legal art. In usual practice, a sole proprietor is also the daily manager of the business, but this need not be true. Nor is it necessarily true that a sole proprietorship has only one person working at the business. It refers to a technical legal relationship. (See Wikipedia for "sole proprietor.")
In the U.S., one term commonly used for people who work for themselves is "self-employed," regardless of the legal form of ownership, which would not normally be known or of significance to most people. That term, however, does not imply that the person works alone. A person who owns and runs a small shop with several employees would still be called "self-employed." A person who works alone may be called a "sole practitioner" or a "lone practitioner" although that term encompasses people who do not work alone so long as those with whom they work are in an auxiliary capacity. Thus a doctor who is not in a group practice would be referred to as a sole practitioner even though the doctor may employ a nurse or secretary or receptionist, none of whom practice medicine.
EDIT: I see the the OP was particularly interested in the incidence of taxation. The entity that has primary liability for taxes depends on the exact legal form under which the business operates. There is, however, a doctrine in U.S law called "piercing the corporate veil" that may impose secondary liability for corporate debts on other parties. Moreover, U.S. tax law is arcane and Byzantine and so is the domain of highly specialized accountants and lawyers. Any questions about primary and secondary liability for taxes should be asked of those specialists. I won't even venture a guess.
The term self-employed without employees seems to be used by multiple organisations: