Need clarity on car rental insurance to be covered

1. AFAIK your situation is always the case.

2. This is true, the promise to pay for damage, and can be true the promise to pay for your health costs.

3. Some liability is always included in a rental since it's the owner of the vehicle, not the driver, who is approached first for damage a vehicle causes, so Enterprise will include basic liability in every one of their rentals. You may be exposed to liability here if the basic coverage included is not enough but who knows. Just because supplemental, or additional, liability insurance is frequently sold doesn't mean it has ever been needed.

are you dealing with Enterprise in US or UK? It's interesting, the UK website makes no mention of being able to opt out of basic insurance. Normally all insurance is included but there is at least £1000 of excess (which can be reduced by £900 on payment of daily additional fee). It would be illegal to drive the car without liability insurance and even if your card covered it you would probably not have suitable documentation to meet legal requirements. What you seem to be saying is that Entreprise are offering a rate with no insurance and then suggesting adding Liability, this is something they might do if renting to a company with a cover all business policy and would be a fair solution in your case. What you need to check is that they aren't quietly leaving the main insurance in the price.

Actually reading the Enterprise website now I can't make a lot of sense out of it either, and I have have used them in the past in the UK. Note that they are calling liability "3rd Party" if that helps, but I can't see well what is and isn't covered, either. If you can't refuse CDW (they use the term DW) then your credit card should not penalize you and suspend coverage, and their coverage would then cover the deductible (called "excess" in Europe) but worth checking whether they follow this policy-- it is standard practice.

"Third party" is what it is called in the UK. It means anybody who is not the person being insured or the insurance company (the first and second parties in the insurance contract).
It is a legal requirement in the UK, and all other European countries, to have third party insurance for any motor vehicle on the public roads. So it is included automatically in the price.

What is not a legal requirement, and so is not included automatically is:
1) Loss or injury to yourself or other passengers in the rented car (theft, damaged property, injuries), that is the renters problem.
2) Damage to the rented car. This is what the hire car company is worried about, and why they want to sell you (C)DW insurance, or hold a very large amount on your credit card.

Important note: Being covered by your credit company for a hired car is a US peculiarity. I know of no UK or other European country where this is the case. Neither do normal car insurance policies cover you for hired cars. Car hire company staff, especially in smaller offices, may not have heard of either of these before and may not believe you.
They are used to their standard procedures, where they either sell you DW insurance (with perhaps an "excess" of £500 or £1000), OR put a large "hold" on your credit card until you return the vehicle intact.

Buy everything they offer. From what I hear and read, if you try to use your cc which does cover some accidents, you have to pay upfront. Then submit bills to cc for them to pay you back. I would rather just pay the insurance upfront and be done with it. Consider it part of the expense of traveling.

I wouldn't worry at all about paying upfront. I think this is one of those needless worries frequently reposted here.

I've never heard of anyone ever using the additional, or supplemental, liability/ third party, the 8 pounds/day you are being offered. Chase may not have understood the situation with their recommendation. What you are protecting yourself from is an international civil lawsuit from someone who has severe injuries or substantial lost wages or wrongful death. Unless you are often a criminal driver, driving drunk or recklessly speeding I just wouldn't worry about it.