I believe that the non-technical person would have a hard time accepting the argument that even though temperatures inside the battery could still reach over 2000F, since there is no open flame there is by definition no fire.
The uncontrolled release of the chemical energy stored in the active materials of the battery is technically not fire, but it is capable of doing all of the damaging things that fire can do.
BTW, charcoal does not burn by that same narrow definition, since there is no open flame once it is started. But it sure does oxidize well and gives off a lot of heat.