Continuation of old direct insurance from 2001

Do I understand your statement correctly that a new contract would have to be concluded when changing employers? It is an ongoing contract that is supposed to be continued, or am I mistaken?

The contract runs through the employer; if that changes, the contract ends because no one can continue it. Whether you are allowed to make private contributions, i.e., from your net salary, must be answered by the insurance company. If the guaranteed interest rate is attractive, it is certainly not a bad investment for RK1.