Liability Auto Insurance while travelling

Liability insurance will be included when you rent the car. It's the collision/comprehensive insurance (damage to the car) that may no be included in the rental.

Liability (third party) insurance is a legal requirement in all European countries. It is therefore included in all rentals.
As Andrew says, It is collision damage to the car itself which is not covered. This is the optional "CDW" coverage. If you don't have this, and damage the car, you have to pay.

Also not covered is injury or loss (theft) to people in the car (your self and passengers), that is your responsibility.

We skipped it when we rented a car in France...BUT...we gave the vehicle a really sound look over before we exited the parking lot with it...found several dings and scratches not listed on the rental form, had to go back to the counter and tell the rental agent. When we returned the car at the airport, the staff barely looked at it and just said "Okay". While driving, we threw our empty food bags/water bottles/whatevers on the back seat to look more like a local (per RS advice). If you aren't a good driver when it comes to stick shift or narrow streets, buy the extra insurance at the rental counter for your peace of mind. Although expensive, it would be far worse, in my humble opinion, to be tense and worried all the time while you are on your vacation!

If I am a local, do I also have to throw old water bottles onto my back seat so that I look like a local? I shall need to buy some especially. Is there a particular brand that is more authentic? I currently have a hat and a road atlas on the back seat. Is that sufficient?

Does Rick Steves still spout this nonsense?

We always buy the extra insurance offered as our car insurance or credit card does not cover international rentals. Check kemwel, have used in the past with the prepay option. When picking up the car , just like here, they review what’s covered and what’s not. Same for Sixt. Check Avis, Hertz but I never found them cheaper. Strange, in some countries bumper to bumper would not cover windows or tires (Spain) so need to add a few extra euros for that! You should shop around on the internet and compare apples with apples. Also check different rental locations for rates.

European governments are really adamant about every car having a "green card" which is proof of insurance. Rental car and leasing companies all have liability coverage.
Some credit cards provide collision and comprehensive coverage in Europe. You just have to get online and pull up the list of terms and services to verify what your benefits are. Chances are optional CDW is not needed if the card you're charging the rental on has the proper terms--except in Ireland and Italy where governments have made optional CDW mandatory.
If you're going to be in France for 3 weeks or longer, there are many factory sponsored car leasing programs that have benefits over week to week car rentals. One is that there are not limitations on where the cars can be taken. Also, anyone in the car is covered on car insurance when driving. Thirdly, a 4 week lease is cheaper than a car rental. See AutoEurope.com or the internet for details.
(I used to manage one of the largest car leasing portfolios in the U.S.)

Curious, the OP wrote "Liability Auto Insurance", which to me and Andrew who posted before me means insurance against damage or injury to anything or anyone outside the car, AKA third party insurance. This is a legal requirement and is always included.

The later posters are talking about insurance against damage to the rented vehicle, AKA CDW, which is not a legal requirement and is an add-on, though a sensible thing to have.

Which did the OP actually mean?

CDW is not needed if the card you're charging the rental on has the proper terms--except in Ireland and Italy where governments have made optional CDW mandatory.

Correct me if I'm wrong, they don't say you have to have it, they just don't accept foreign (non-EU) insurance, so your card insurance isn't valid. So you either buy the CDW, or have a policy with a local company, or leave yourself exposed
Note having insurance on credit cards is a US peculiarity, you don't get it in European countries.

Fyi- current yellow vest protests about additional taxes in France for diesel fuel, which the majority of cars use. This most likely will affect your rental. You go to the counter and they add this and that. Or personal injury is not covered, or this airport tax, or city tax. Just saying

@Chris
You invited us to correct you if you are wrong, so I’m accepting your invitation. We always use the Amex Premium rental car insurance (>$30 for a period up to 42 days) to cover damage to the car, except in Italy where we rely on our Chase card. We have had 2 occasions to make claims on the Amex, and one on the Chase. All were handled promptly and without hassle. I’m sure there are variations from card to card and country to country, so it behooves one to check with the card issuer. Amex insurance, e.g., doesn’t cover Italy. However, it is simply incorrect to state that one must buy the rental company’s insurance or risk financial liability.

As has been said, every car comes with liability insurance so you can't buy a separate policy.

Collision damage waiver (CDW) is the kind people buy. I use my Amex card insurance because it's only a one-time $25 for each rental (up to 40 days). I had my first accident in Carcassonne in June and, with some work on my part, they paid as agreed. Amex insurance is subject to limitations by different states in the US and by some EU countries. It's worth looking into but, like all insurance, pays to know the details.

I have had two minor scrapes on rental cars while in France. Both times my Visa Signature card covered them, and I never heard anything further about them