Yes, yes, yes to everything said here.
The Airbnb liability coverage (“Host Protection”) is a bit better than the property damage coverage ("Host Guarantee). For the liability coverage, Airbnb is the primary insured, the host is an additional insured party. So it’s actual insurance.
Airbnb says its liability insurance is generally (weasel word alert!) primary, in other words it pays first before your other insurance, and covers actions against you.
Here is article by Proper Insurance on the shortcomings of both types of Airbnb coverage (of course coming from the perspective that they want to sell you insurance!)
So I have private STR insurance AND a liability umbrella.