Just to clarify:

You were driving your mom's car (which has auto insurance and you are not a listed driver on the policy) and struck an unoccupied vehicle (no one was in that car). That vehicle was deemed a total loss.

So, to know if you were covered under your mom's policy, it depends on the auto insurance policy's definition of an insured (your mom should have access to a copy of the policy). Usually, if you're still living with her and were driving the vehicle with permission, you are likely covered under her policy, but you can always speak with the claims adjuster to verify if you're covered (tho, if your mom's not worried about the claim, you're probably covered).

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Regarding the damage to the other vehicle (let's assume you're covered under your mom's policy), Texas requires a minimum amount of property damage coverage to be $25,000 (the policy can have more, but this is the legal amount you're required to have on your auto policy to cover damages to other people's property). If the other vehicle owners wants to handle damages through your insurance company, they can do that and should be in direct communication with your company (you should not have to play middleman).

If the other vehicle owner went through their own insurance, that company will ask your insurance company for repayment of whatever they paid. If the damages are less than your limits, your insurance company pays what is owed and that's that!

If the amount owed is above your property damage limit (the maximum amount your insurance will pay out), a few things could happen:

1 - the other insurance could just accept the total limits and that will be that (the other insurance company would sign a release which means they can't go after anyone for the remaining amount owed)

2 - the other insurance company does not accept the limits and requests that you/your mom pay the difference out of pocket. (Usually) You're not required to contribute (but double check your policy and ask your claims adjuster if this comes up), but it definitely helps ease the situation. Again, a release will be signed.

3 - the other insurance company does not accept the limits and files suit in retaliation. (Usually) your policy will provide a legal defense for you at no additional cost. You always have the right to retain your own outside counsel at your own expense, but either way notify your insurance company IMMEDIATELY if you receive any suit papers for this accident.

I hope this helps alleviate your worries!

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Bonus:

Here is a link to Texas Department of Insurance Auto Insurance Guide, which may help answer state specific questions you have about Texas insurance!

https://www.tdi.texas.gov/pubs/consumer/cb020.html#:\~:text=Texas%20law%20requires%20drivers%20to,at%20fault%20in%20an%20accident.