That sort of behavior makes me think this lady either has a serious mental illness all her life (if she was always like that) or some dementia type disorder like Alzheimers.
A lot of times people who live alone have serious issues with dementia, etc and if the have adult children, the kids are in denial about Mom OR they just don't see her enough to know how bad it is.
I think you did quite well under the circumstances - she obviously has a mental problem - whether dementia or just all her life. Your first priority is to protect YOUR family. And thankfully the neighbor was there to help out.
Let's see, you have a person raving about killing your dog, pushing their way inside your home and holding a potentially lethal instrument. If she had been a 20 yr old male instead of an old lady, you'd have called the police in an instant, right??
So, you did the right thing to close the door on her - she needs help - that was clearly irrational behavior on her part - CLEARLY.... as far as calling the police - I think that's just your call.
If you want to go one step further, instead of offering flowers, or instead of calling the police, call your local agency that deals with elderly - probably Social Services, or whatever the agency in your community that investigates child neglect if you can't find the right one - they should be able to tell you who to call. She needs some outside intervention, in my opinion. I'd tell them the story, and explain that you don't want to get her in trouble, you want to get her help before she harms herself or someone else, or someone's pet.