Your car is insured. Period.
It does not matter who's fault it is.
This is factually wrong. Period.
There are many different types of auto insurance.
The only one legally required in most states is liability- They pay for OTHER peoples cars/injuries (and only if it's YOUR fault in fault states). Not YOUR car or injuries.
So no, your insurance company would NOT fix your car if you only have liability coverage. Which again is generally the only coverage legally required.
There is no such thing as an insurance company saying - "You are only covered if its not your fault".
That is literally what they will tell you about covering your own car if you have a liability only policy (and then they'll tell you if it's NOT your fault you need to call the OTHER guys insurance company to get paid)
Auto Liability Insurance: What Is It And How Much Do You Need? - Cover
Liability insurance covers the costs of injuries and damage you could do to other people and their property when driving. However, this doesn’t cover the driver or the driver’s property.
The insurance that covers YOUR car (especially if it's YOUR fault) is called collision. The insurance that covers your car for things like fire/theft/glass damage is called comprehensive. There's also separate coverage for uninsured drivers. And for medical payments. These are all separate parts/types of car insurance- and none of them are required generally.
(disclaimer- if you have a loan on your car, the leinholder likely DOES require full coverage while they hold the title to protect their interests)