It "really" depends on how you define "modern," which has no single, accepted definition, especially with regards to war. It is not an exceptionally meaningful or useful term. The real question to ask is: why does one care which is the first "modern" war? What are the stakes in picking one war over another? Because it is in those kinds of details that one will find more concrete answers to this question.

As it is, it is about as meaningful a question as asking which war was the "worst" or the "best."