As long as "you" are buying quality.
There are a lot of junk pits out there and I would not risk my retirement home.

We have had one in our screened in lanai with 10’ ceilings for the last 3 years. No issues and it makes the lanai usable especially on cool nights.

Our fire pit is outside on our patio and we love it there. It is nice to sit out and enjoy a fire, look at the stars and moon etc. we did not want to deal with any possible discoloration by having the fire pit or bbq in the lanai.

As a retired insurance claims manager, you may have a denied insurance claim from a fire loss in this situation - I wouldn’t do it.

The safest bet is not under lanai

A lot of variables that need to be adressed, such as size of pit, ventiation on the lanai, locaction of the pit, type of ceiling in the lanai.

Interesting how people who have no problem driving an automobile 55+ mph on two lane county roads with no shoulders have issues with low risks...

If you decide to put a fire pit or grill in an enclosed area (even a screened one) then spend a few dollars to buy a CO alarm with a digital display of the measurement. Just like the one you should have inside your home. It may read zero for a long time. The one day something with the combustion goes bad, it will alert to save your life. Also keep them away from walls and screens. I've seen lots of melted vinyl siding from grills as far as 3ft away.

Don't let the naysayers and overly cautious people dissuade you. Some folks are scared of their own shadows. Use common sense. If you are far enough away from combustibles, and have sufficient height clearance and ventilation, you will be just fine 99 times out of 100. For reference, here's the set up I have on the lake house up north, used often and without any issue. That's a vinyl siding ceiling, FYI. It's a little more spendy than the firepit tables from Academy Sports, but this Bali I got from amazon does a very nice job. That's a standard BBQ grill propane tank in the storage underneath that fuels the magic.

https://i.imgur.com/9tvhjS5.jpg

Restaurants do it all the time.

Fire pit
 

A fire pit that is UL approved and/or approved for LpG should be fine either. However, things to consider in a Lanai is that do you have an enclosed Lanai or ability to shut windows? If so then you must consider the Carbon Monoxide build up with using the fire pit inside the Lanai. Also if there is a leak at the propane tank valve-Propane is heavier than air so it sinks and would be a problem for pets ...and of course any sparks from equipment etc...e.g., don't vacuum your lanai if you think there is a slow leak at the propane valve. You should be able to smell a leak as there is a sulfur smell compound that is added to the propane for detection. If in doubt as to a leak get down on the ground and sniff.
If you suspect there’s a propane leak in your home or lanai, follow these safety guidelines:

Extinguish all flames and sparks in the area.
Get everyone out of the area.
Shut off the gas supply value on your propane tank.
Report the leak to your propane supplier or take the tank to a retailer.
Remain away from the area until your supplier can check the entire system if tank not removable.

Also just a note of caution that if you leave your propane tank out in the hot summer Florida sun it will expand. Propane boils at -44 degrees fahreheit in the liquid form and when it gets hot it boils into a vapor. In the tank in the hot Florida sun the pressure increases in the enclosed space and the propane remains a liquid but wants to boil. If your tank is "old" you might want to have it checked for hydrostatic pressure. (or just swap it out using Blue Rhino).

Had one in our birdcage for years no issues.
Not sorry for being direct on this.

Check with the fire Marshall.
 

Check with the Fire Marshal. He states fire should be 20’ from house and property line.

Weird, in my state they want actual wood burning open-air fire pits to be at least 10 feet from the house. 20 feet with a non-smoke or spark/ember generating fire seems pretty, ummm, overly-cautious. May as well wear a seat belt when you are sitting in your car in the garage listening to the radio also. Or wear a bike helmet when on the couch watching TV.

Not sure putting a co alarm outdoors will really do much. The lanais are open (screened) on three sides aren't they?