If you you wish to exclude those Asian conflicts from The Cold War its OK with me.  But then you'll need to recognize there's also a classification titled The Asian Cold War.  Also a plethora of reports under that title.  

And don't forget the Cuban stuff  -doesn't neatly fit within the European or Asian cold war groups.  Or some conflicts in South America, and Africa centered on whether communism was getting some footholds.

And you'll also see there's been discussions about the cold war having many hot shooting conflicts, so its somewhat misnamed.  And that applies to either versions of how to classify those cold wars.  A broader playing field than taught in grade school.

HOWEVER: Seems to be useless details to continue debating such minor classifications & distinctions.  De minimus