Re: Definitions of employed/self-employed ?
"Employed" - you work for someone else in "their" company (either full time or part time)
"Self-employed" - you work for "your own" company (even if providing services to someone elses company)
No. Self-employed is a fairly specific thing however much the HMRC try to muddy the waters.
If you're self-employed you can provide a service or trade goods and the money that the customer pays is your personal money. You may, however, have liabilities such as tax and NI and the ability to set your expenses against the income before taxes are levied. You'll also have liabilities for any other costs you incur and those liabilities fall directly on you, personally. You are the business; there's no distinction. You could, for instance, buy a bucket, a ladder and some cleaning materials and set up as a self employed window-cleaner tomorrow. If nobody wants their windows cleaned or you go on holiday or fall sick, tough. There's no money coming in and your only fall-back is on the dole under whatever name it's going ATM.
What I've just described is a sole trader. Another variation is a partnership. You can go into partnership with someone else with two buckets and two ladders as a window cleaning partnership. Much the same applies. Any money you take becomes your joint property as partners and it's up to you to divvy it up but you still have personal liabilities. In fact those liabilities are on both of you for the actions of either - your partner screws up and you could still be liable for whatever it costs.
An employee doesn't receive money that's paid for the whatever the business delivers. The business, whether it be a sole trader, a partnership, a limited company, a PLC, a charity or a public sector receives the money and pays the employee. The employee will have to pay income tax and NI although the NI rates are different and there's also an employer's NI to be paid. The employer will also have to find sick pay and holiday pay. The business and the employee are two separate entities. One thing that should be made clear here is that if you're in business as a self-employed window cleaner the your customers are just that, customers; they're not your employers.
There's no real obstacle to a company engaging a specialist worker (to use a generic term) on a self-employed basis. There is, however, a risk and that's a ruling some time ago that if the sole trader were to default on tax payments the IR (as it then was) could look to the engager (again, to use a generic term) to make good. This, AIUI, was what brought the limited company freelancer to the fore; clients were more likely to be comfortable dealing with a limited company rather than a sole-trader as the ruling did not apply to a worker engaged on that basis. It's not essential to structure the arrangement through a limited company. I came across a client who, I discovered, engaged graphic designers as sole traders and they actually had their own form contract for this.
It should be clear from the above that when a freelancer works through a limited company they are not self-employed. It is the company that is the business that receives payment from the client. The company has its own tax liabilities such as corporation tax and employees NI and its own rules about expenses. It's up to the company's management and ownership (who are probably identical with the employee(s)) to decide what payments it makes to its employee(s), subject, of course, to any limits on what it can afford and company legislation about solvency. It also needs to distribute salary payments through sickness, holidays and slack periods. As a company, of course, the business is subject to a good deal more legislation than the sole trader. It needs to go through the necessary legal formalities to be incorporated and it needs to produce the appropriate annual accounts to submit to Companies House. It has more freedom to distribute income as dividends, which have their own taxation arrangements, as well as salary and to retain some of its income when things are going well in order to be able to make payments when they're not. It is also the company which assumes any liabilities and not the employees.
For a small company the same person might be shareholder, management and employee. Nevertheless they are separate roles unlike the sole trader where the person is the business.
NB Occupations which are regulated create exceptions. Anyone can set up as a self-employed window cleaner but not as a self-employed medical practitioner unless they have the appropriate qualifications and registrations. OTOH there may be instances where the professional is obliged to bear personal liability even if they are an employee although the employer may pay for professional liability insurance.