I was looking for the correct formal and legal term that describes a company/firm with one employee.
I came across different terms: one-man business, one-person enterprise, sole proprietor--of which I was not able to determine the most appropriate one and the difference between these terms.
In the UK, the legal term is "sole trader", (see https://www.gov.uk/set-up-sole-trader). This is one of the classes of company under tax law (the others are a "partnership", a "private limited company" and a "public limited company").
A sole trader can employ any number of employees, but they have no stake in the business. A typical example would be an independent plumber.
(It would be possible to set up a limited company with one director who is also the only shareholder, but this would be unusual.)
I believe the legal US term is "sole proprietorship"
You would normally only use "sole trader" in a legal context. In casual contexts you could use any of the various examples you give in the question
If you are earning more than £1000 by selling hairbows on etsy, you should set up as a "sole trader" and register for tax self-assessment, otherwise you could find yourself arrested for tax evasion.
I heard Mary's one-woman business is doing really well, she made £8000 pounds last year selling hairbows!
