I think gig-employed = self employed but applied mainly to those at the bottom of the wage scale doing delivery work or similar, like food delivery services, couriers etc. They are usually paid a set amount per delivery/trip/work task.
Zero-hours are employed by a company, often retailers like supermarkets or courier companies but the employees contractual working hours per week are set at zero and they can be asked to work as many or as few hours as their employers require - and only get paid for the hours they work.
With both the above weekly wages and working hours can vary widely from week to week.
Generally, self employed means you work for yourself and choose how much or little you work, what you do and your charges, bearing in mind the industry you are in and what is considered the going rate for the job. The Inland Revenue rule used to be that you were not entirely dependent on one employer for all your work (must have changed, see gig economy above.)
Dh is a retired Chartered Engineer and is self employed. He only works for one employer, a friend, and only takes on the work he wants to do, he is entirely free to turn work down - and does.
This is a first run at a definition and I am sure others can improve on it