Is this a fallacy?

It is just factually wrong. First, war only sometimes causes those things, which is enough to make war and apocalypse distinct. Second, the justification is a non sequitur . That which includes X is distinct from that which causes X. The flu includes nausea and headache. My wife is not the flu (ha). 

It seems that Person 2 ignores scales. This is a false equivalence.

Example: The reason why you go to the movies is to watch a movie. You can do that with your own TV. Therefore, you don't need to go to the movies. 

I would also add that Person 1 includes 'war' as separate from the other things mentioned to prove what he/she wants to prove. This is begging the question because the distinction needs to be made.

Example:  Supermarket A has more things than supermarket B because supermarket A includes everything that supermarket B has and also includes the electronics section.

In the example, this begs the question if supermarket B doesn't have an electronics section. 

Notes: I've noticed that I don't generalize this logic problems in terms of cognitive biases. Instead, I focus on specifics by thinking of examples. Perhaps because I want to classify things as thoroughly as possible. With this said, I'll attempt to give a generalized view of thinking gone wrong: Person 2 might be motivated to make a conclusion and thus avoids the issue. This would be avoiding the issue.  Person 1 perhaps wants to make a distinction because they already want that distinction. Maybe they don't want to elaborate or something. This would be wishful thinking.