I have never even considered requiring tenants to carry renters insurance because I have a landlord and umbrella policy myself.

When I was trying to get details about having an umbrella LLC, so I could have some continuity, but not share the liability across all properties, he mentioned having a property management LLC as my umbrella. 

Will the insurance be under the LLC name or I can keep the insurance and umbrella policy under our personal name. 

Make sure all railings/ gutters are secure, no crumbling masonry ect.You can also opt to add a personal umbrella if you are going to hold the property in your name for excess liability protection.GoodLuck!!!

Doing it again, I would have started with buying properties in my own name for the better financing options and I would have bought an umbrella policy to cover me.  

You should have adequate insurance including a high limit umbrella policy, as a first step.

Right now we have a commercial umbrella policy in lieu of using LLCs to hold title.

Should I move this rental (and others) into an LLC or buy an Umbrella Insurance Policy on it?

You will need an umbrella to cover most anything you do.

I have 3 Rentals that are all housed in their own seperate LLC's.I would like to get a Umbrella Policy as well.Spoke with a couple Insurance Agents and finding it difficult to find a carrier to provide an umbrella that covers me personally due to houses sitting inside LLC (Mixed use).