Your confusion is quite common and it's great that you're taking the time to understand your coverage before you go on your city breaks.
Your health insurance seems to cover you for inpatient care related to accidents and emergencies abroad, as well as elective procedures abroad once you've had them pre-approved, which is fantastic and contributes to your safety while on travel.
The European Health Insurance Card or EHIC allows you to access medically necessary, state-provided healthcare during a temporary stay in another EU/EEA country for free or at a reduced cost. It primarily covers you for the treatment that is needed to allow you to continue your stay until your planned return. It also covers you for treatment of pre-existing medical conditions and for routine maternity care, as long as you're not going abroad to give birth.
However, neither your current health insurance coverage nor your EHIC replaces travel insurance.
Travel insurance typically provides coverage for the non-health related aspects of your trip including:
1. Trip Cancellation or Interruption: In case you need to cancel your trip for a covered reason like illness, injury, or death of a loved one.
2. Baggage & Personal Items Loss: In case your luggage or personal belongings are lost, damaged or stolen.
3. Flight Delay or Cancelation: Coverage for any additional expenses such as accommodations and meals in case your flight is delayed or canceled.
4. Emergency Medical Evacuation: If you need to be evacuated due to a serious illness or injury, your travel insurance would cover the costs, which can be substantial.
5. Repatriation of Remains: This covers the cost of returning your body home if you were to pass away while abroad.
So while your healthcare coverage and EHIC are very important and helpful, they have their limitations. Of course, whether you should get extra travel insurance would depend on your comfort level and risk tolerance. But it's always better to consider all the eventualities and be financially protected. After all, unexpected incidents can occur and can turn out to be quite expensive if one isn't properly insured.
I hope this answers your question! Enjoy your city breaks!