You should check the Texas Department of Public Safety to find out for certain if the police could cite your daughter for driving without insurance in this instance because she showed your insurance card instead of one that had her name on it or not. If may be that the police could if he was being a stickler for the letter of the law but likely if your daughter came to court and showed your proof of insurance for the car and then whatever car insurance she has in her own name that the ticket could be dismissed. If you are the owner or operator you must show insurance but the vehicle must have insurance on it when you take a test. If you are the registered owner of a car you must show proof of registration and liability insurance when you get a TX driver's license.
In Texas all applicants for a Texas driver's license must show proof of insurance to get a driver's license. You have to show proof of insurance regardless of whether you are getting a license for the first time, renewing your driver's license, or getting a Texas driver's license after moving from another state. So in order to be licensed your daughter needs to have auto insurance that covers the car. If she lives in your household then she would need to be on your policy normally. If she lives outside of your household then she would need insurance on her own car or a non-owners policy if she does not have a car to insure.
The Texas DPS and Department of Insurance note that you have to show proof of insurance any time a law enforcement officer asks you to. A law enforcement officer will ask you for proof of insurance if you are involved in an accident, regardless of whether you caused the accident or not. Police will also ask you for proof of insurance if you are stopped because of a traffic violation or for any other reason. You also have to show proof of insurance when you get or renew your driver's license, register your car, and get your car inspected.
You are correct that people allow others to borrow their vehicles all the time and usually they show the car owner's insurance card, again we would advise you to check with the TX DPS to see if a car driver must also carry their insurance card on them when driving another person's vehicle.
In case you were wondering in most cases when you allow someone else to borrow your car, your (the vehicle owner's) insurance pays for losses, regardless of who was driving. This means that if your daughter was in wreck while driving your car, your policy will usually pay. If your daughter has her own auto policy then typically her insurance would pay any remaining amount that exceeds your policy limits.