However, as the page says, you're personally liable if something went seriously wrong.
The alternative is to pay £50 to register as a ltd company, though there are costs in running one (£12.50 per year + doing your own accounts vs paying an accountant £200-£300).
I think this is called a "trade name" in other countries. Wikipedia has some examples for other countries.
Paypal could be a good way to take money from people privately, and it's probably entirely legal, up to a certain point.
But you still have to register some form of sole-proprietorship company?
I think you can have problems with payment processing companies as they might not want to make business with a private person.
Of course there are downsides, you will look a lot less professional without a KvK or Chambers of Commerce number.
But if you want to look professional, you need to incorporate