I have recently qualified as a level 2 fitness instructor and I've heard that having personal liability insurance would allow me to act as a level 3 personal trainer would, with regards to teaching gym-based activity. Is this true?
If you are performing duties on behalf of the gym where you work, then you are covered by their insurance. You can instruct in the gym, develop training plans, supervise or motivate the customers and consult 1-on-1 on their exercise preferences. If your employer chooses to do so, they can call you a 'Personal Trainer' or asign 'Personal Training' duties to you during some of your working day. You don't need ANY qualifications, at ANY level. However, it is normal practice for employers to insist that their employees hold some certificates; which certification is appropriate to each level of employed role is entirely determined by company policy (if they have one).
If you are NOT performing 'personal trainer' duties on behalf of your employer (or during time paid for by your employer, which legally amounts to the same thing), you can get insurance to cover the company that you ARE working for at those times, i.e. yourself.
These insurance policies would probably insist (or the insurer would insist if you make a claim) that you need to be registered as self-employed for tax purposes, with an official business address (not necessarily where you carry out your training operations), and 'suitable' qualifications & professional memberships for the type of business you are carrying out.
So if you are a member of a professional trainers' association, the organisation will offer insurance (often as part of the membership) that is suitable to cover you for training. Usually, membership will depend on holding a qualification; Level 3 is a typical minimum required by these bodies.
If you are not a member, the available insurance may cover you for the kind of activity you want to carry out, but you may have a hard time getting the insurer to say that it does, in a courtroom.