My mom and sister would regularly read Ann Landers advice column. She is long gone but my advice to them applies. People offer you advice but in the end if it goes bad the same people will say sorry to hear that.

My view, there is right, there is wrong and there is legal, they are often not the same.
You know what the right thing to do is. That tree could fall at the wrong time and injure not just property but you or your neighbor. Which way will a tree fall is easy to tell. It is common for a tree to lean and it will most likely fall the way it is leaning. Like Ann Landers I accept no responsibility should it go bad.

Re: insurance
There is usually a deductible

Re: people
My guess is the original poster knows what the right thing to do is.

Under Florida tree law, the health of the tree determines who is responsible when that tree causes damage to a neighboring property. Gallo, 512 So. 2d at 216. As long as the tree was healthy at the time, a landowner is not liable for damage caused if the tree falls down onto a neighboring property.Mar 17, 2021

Do the right thing and take the tree down.

The neighbor knows because we mentioned it to them.
 

The arborist who looked at the tree said it was lightning that did it. The tree still leaves out put you can see part of the tree roots moving in the wind. We will have it removed but it's in a tough spot. No access for heavy equipment and you would have to climb it to top it before trying to drop it in the yard because of its size and proximity to your house/deck.
Thanks for all your thoughts on this.

I’ve been in property management for 33 yrs in Colorado. We manage rentals & HOAs. When this happened at one of the rentals (house hit by a downed branch from the neighboring property), the insurance company told us the damaged property’s insurance pays & they work it out with the other insurance. I know it sounds unfair & I had difficulty accepting it but that’s what they said & did.

Another thing to consider is what the DNR and/or Corps of Engineers and/or the Coast Guard would say about letting a tree fall off your property into a waterway. (Don't know the size of the lake, where it is, etc.)

Court will frown upon someone not remediating a known problem. Good luck denying this post in discovery. Just fix the problem

An insurance agent in Georgia explained to me this way: If your tree falls on your neighbor's property it is covered by your neighbor's insurance. If the tree is dead then it is covered by your insurance. In order to be liable you must be negligent.

I don't think so legally anyway. We had a neighbor's tree come down on top of our house during a hurricane and our insurance company told me it was our problem to get it removed and fix the damages, not the neighbors. In this case, your neighbor may be able to shift the onus onto you if they notify you in writing of the impending danger to their property but I am not a lawyer or insurance agent.

Tight Wad
 

Holy cow the tree is on your property-take it down. I would be very disappointed to have a neighbor that doesn’t take matter seriously and remove the tree because of cost.

My thoughts also.

How would a fallen tree sink to the bottom of the lake as a fish crip,
wouldn't a dead tree float?

have it taken down on the ice in the winter and it can be easily cut up and stacked on the shore for removal in the spring

Was in the insurance adjusting business and agree with most of your thoughts. Many people use the term "act of god". Will not find that term in the law. Over the years people have used it to refer to weather events.

In our litigious society if the tree hit a person and killed/badly injured that person you can or should expect that the person will try to find blame on the tree owner. There are a lot of "paid experts" out there who will make an arguement that the tree was likely to fall or the owner of tree knew or should have known that it could fall. If something did happen, if you have a Homeowners policy with liability insurance. they would respond to any claim and/or lawsuit.

Was in the adjusting business. Most policies are called all risk policies. In other words the policy pays for all types of damages caused, EXCEPT for certain causes(such as war, wearing out, flood) In Florida, your insurance company could try to recover the money paid out if they could show that the neighbor had a bad tree and knew they had a bad tree