More or less. Batteries store electricity with a reversible chemical reaction, and one of those reactions is oxidation. It isn't exactly true to say a battery generates electricity by burning its cathode, but because it requires elements that can readily oxidize with a lot of energy, it's not tough for that oxidation to turn into fire if something goes wrong.

Just because your phone hasn't caught fire doesn't mean it can't cause fire. If it's a lithium battery then, by design, it is more of an incendiary than other types of batteries. Lithium batteries provide their own fuel, oxygen (it makes it) and energy. That's why all lithium batteries (not just cars) are hard to put out.

I imagine you have criteria for what constitutes when it can become one, but I don't think your phone battery will survive a car accident and then the subsequent burning of hundreds of other batteries burning right beside it. AFAIK, there are no protections for lithium batteries that make them fail safe, which is why there are restrictions on where they can be used or transported.