The issue of paying for training is interesting but it's a sideshow.
In the UK, the relevant test is the "IR35" test. If it walks like a duck, talks like a duck etc. Written contracts, side letters, are all irrelevant if HMRC determines that you are (in effect) performing the duties of an employee.
It has to be said that HMRC enforcement of this is patchy (to be polite) given that many senior figures in prominent roles (eg at the BBC) have been playing this game for years.
Personally, having been subject to a HMRC investigation (because of the actions of a former employer in the motor industry) I would never ever want to be on the wrong side of HMRC.
Read more here
https://www.gov.uk/guidance/ir35-find-out-if-it-applies