Or should we buy it under our names and purchase insurance policy + umbrella policy?

Insurance would cover it, I would pay the deductible, and no assets would be lost.If you are in an area like San Diego where people are more likely to sue, a judge is more likely to find you guilty, and the payout is likely to be higher, then you may consider an umbrella insurance policy.

LLC - Would you say it's smart to have an extra business insurance policy (like a personal umbrella,) on top of your standard Property Liability Insurance to cover being sued for accidents?

Form an LLC and if you are concerned about liability can also purchase umbrella policy in addition to your property ins.

I personally decided to create a holding company in Wyoming (which functions as an umbrella LLC).

We hope to use this LLC as our "umbrella LLC" for our real estate investment portfolio going forward to be able to utilize a business line of credit rehab and intital investment costs. 

I have umbrella coverage but no LLC. 

I think that there is a case to be made to keep properties in your own name for easier financing, and have an umbrella insurance policy over the properties to handle liability. 

The declaration page shows personal name and LLC listed.Now I am trying to add them in Umbrella policy but the agent says - Rental properties would not be included under the personal umbrella because they are insured as LLC.What is my best option ?

I was also looking into an umbrella insurance policy.