The simple differences are that, “finished contract” means that the helper completes the full two-year contract, “break contract” means that the helper quit the job earlier, and “terminated” means that the employer fired the helper.
To the Immigration Department, the distinction between these three types also has huge significance. A “finished contract” helper can apply for an extension of stay visa and delay her annual leave at the end of the contract, and “finished contract” Filipino worker can also deal with the hiring process herself without using an agency. “Break contract” or “terminated” helpers can have a record in the Immigration Department, and they might be rejected a visa in Hong Kong if they have broken their contracts or have been terminated for multiple times.
For detailed information about the differences, read here.