Returning to the topic of what a refugee is, this is what the Handbook on Procedures and Criteria for Determining Refugee Status under the 1951 Convention and the 1967 Protocol relating to the Status of Refugees says:

CHAPTER V – SPECIAL CASES

A. War refugees

164. Persons compelled to leave their country of origin as a result of international or national armed conflicts are not normally considered refugees under the 1951 Convention or 1967 Protocol.
In other words, coming from a war zone alone is not enough to qualify for refugee status under these papers.