"The building is insured, so I am covered"
The landlord's policy covers the structure and the landlord's liability. It covers none of your belongings and none of your liability. If a neighboring unit floods yours, their cover answers for their property, not for what you lost.
"Water damage is water damage"
A burst supply line upstairs is usually covered. Surface water that came in off the street in a monsoon storm is not, on any renters form. In a city where rain arrives two inches at a time, the distinction decides the claim.
"Liability is the landlord’s problem"
Not when it is your dog, your guest, or your cooking. Renters liability follows you, and in a dense rental market the person downstairs is close enough to make that real.