You are correct.
In a legal suit, if you have a "Beware of Dog" sign on your property and someone is bit, you are at fault.
Posting a sign is an admission before the fact.
By law, no dog can bite anyone, anywhere.
They use to put dogs in department stores at night to keep burglars at bay and no longer can, because of lawsuits.
Bad guys got hurt while stealing.
By Texas law,(called the Castle Rule) I can shoot someone from entering my home at night and not be held accountable, due to self defense, but, if my dog bites the bad guy before he gets into my home, I can be liable in a law suit.
In the county were I live, if someone is dog bit and goes to a doctor to be treated, a sheriff report is made. The sheriff goes to the home owner and serves a vicious dog warning. If that dog ever bites anyone again, the sheriff comes and takes the dog away, to be put down.
I have a barking dog, for this reason.
I will do the biting.