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The Warranty company do not want to pay your claim no matter if you buy at a new car dealer or if you buy from a buy here pay here car lot. The main obstacle is having them fix everything for free. Some may mention bumper to bumper but that is not always so. The third party warranty companies will ask you what happen to the car. Your best answer is to say I do not know, I just pulled over and called the tow.
Giving any details about the vehicle breakdown is acknowledging you knew the problem but continued to operate the vehicle. You can also say that the car just stopped while you was driving then you call the tow guy. Be sure to only mention these words because if you deviate something else your warranty can be labelled as void when you buy from a buy here pay here in Florida.