What is the very nuanced and technical difference between Tenant Liability coverage and Damages to Premises Rented?

Dealing with simultaneous situations where I can save money as a Tenant switching from Tenant liability to Damages to premises rent but the Lease calls for Tenant Liability, wouldn't want to be in breach and broker keeps telling me it is the same thing and also renting out my own commercial space and trying to figure which version should be requested of the Tenant.

I understand the Tenant Liability could be broader and include some aspect of medical or injury but might depend on carrier. Any explanation or suggestion is appreciated.