If you pay yourself a salary from a ltd company there's ~14% Employers NI, 12%-2% Employee NI Depending on tax band, 20-40% Income tax depending on tax band. If you pay a dividend there's 19% Corporation tax plus 7.5%-32.5% Dividend tax.
So 46%-56% tax on salary and 26.5%-51.5% tax on dividends. Nowhere near 13%, unless you're talking about 'creative accounting' by running non-business expenses through the company to effectively get them tax free.
I don't live in UK anymore and I could do something similar in my location, but I want to pay taxes and my effective rate is good.
I'll have to enquire with the few mates there -- don't see a reason as to why they'd bullshit me. Frankly I have nfi however, apologies for the cop-out of an answer.
Sole traders are different to one-person Ltd companies. As a one-person Ltd, you used to get a lot of tax breaks, but over the last 5-6 years they seem to have been steadily eroded.
Anyone paying a rate as low as 13% now is probably using a borderline-legal grey-area scheme, which is dangerous because HMRC not only hit those folks up for back taxes when they find them, but put fines on top too.
Though you'll get to higher tax brackets with that pretty quickly too, so will likely want to leave a fair share of revenue in the company (you still need to pay 20% corp tax on profit), or put it into your pension (tax relief on up to £40k/y).