When I first started part time instructing I got myself an accountant and he told me I needed to register as self emplyed with HMRC and pay NI employment contributions on my earnings as an FI. These came to about £120a year.
Having done my tax return for my second year as a part time FI (working for more than one Flying School) I was advised by my accountant that as my earnings as an FI were below a threshold (which was about £4500 per annum) I did not need to be paying NI contributions on my earnings as an FI! I had pointlessly paid about £250 worth of NI contributions, or about the cost of an FIE fee for my first FI renewal.
I was given a form to complete, which I sent to HMRC, and they wrote back saying I no longer needed to pay NI contributions.
Perhaps the area that is causing the discussion is FI's who are paid some sort of "retainer" by the school they work for? I know some schools do this and it might muddy the water if you claim to be "self emplyed".
As for me personally I have no contract of employment with any of the schools I Instruct at, I have no retainer. I fly when the schools have work and when I want too (usually the same thing though!). I am not allowed to charge for Groundschool. I only get paid once the engine starts and I pay for all my own medicals and renewals. Sounds like self employed to me!