This seems like needless pedantry at least for a conversation outside of school. The USSR (with whom the Cold War was being fought) was dissolved and Germany was reunified. For all intents and purposes it did end, just a new conflict (now with Russia) began.
Seeing as the Cold War was a framework for conflicts, not a conflict in itself, I find it extremely illuminating to consider the view of the opposing party. Not a technicality in the slightest.