What is war without direct military conflict?

Technically, "Peace".

But you're probably looking for the term "cold war", which was actually an arms race between two sets of atomic weapon armed nations; which never fought each other with their military forces (there never was a WWIII).

You really cannot use the term "cold war" to describe any "war without direct military conflict" because then all peaceful times thru out history could be called "cold wars." This particular cold war (1945-1990) was strictly between the nuclear armed communist nations and the free world, led by the US.