the short answer
Online personal trainer insurance is liability cover that protects a remote coach if a client claims their programming or advice caused injury or financial loss. Most online trainers carry professional (civil) liability as the core policy, and add public liability for any in-person work. Premiums vary by country, cover limits, and claims history.
Yes - most online personal trainers and online coaches should carry insurance. You are still prescribing exercise and nutrition that a real person performs, often unsupervised and sometimes with a health history you never fully see. Online personal trainer insurance, usually a mix of professional (civil) liability and public liability cover, protects your business and your income if a claim ever lands. The good news: smart screening, signed agreements, and staying in your scope make it far less likely you ever need to use it.
This is general information, not legal or insurance advice. Insurance requirements vary by country and state - check your local rules and speak to a licensed broker.