I explain to clients that they are considered the first party, while their customers are regarded as the third party. First-party coverage protects your business from losses resulting from a cyberattack. Third-party coverage, on the other hand, protects your customers.
If a cyberattack compromises sensitive information related to your clients, third-party coverage protects their risk. If they sue you as a result of the breach, third-party coverage also protects your business in the lawsuit.
When considering insurance options, if the word "customer" comes to mind, third-party coverage is likely what you need for whatever scenario you imagine. If "company" comes to mind and words like "operational" play into the scenario, first-party coverage is likely the most appropriate option.