What does being a sole trader mean in practice?
It is the simplest way to be in business. You register with HMRC for Self Assessment, and that is the whole of the setup. There is no company to form, no Companies House filing, and no public record of your accounts. You can trade under your own name or a business name.
The business and you are the same legal person. Its income is your income, taxed through Self Assessment along with anything else you earn. Its debts are your debts, which is the part that matters if a job goes badly wrong — there is no limited liability shielding your house.
Money you take out is called drawings, not wages, and it is not an expense. You are taxed on the profit the business makes, whether you took it out or left it in the account. That single fact explains most of the confusion new sole traders have about their tax bill.