Just "The War" usually.
To the OP, might I suggest you refer to it as "Saigon", not Ho Chi Minh city. You'll get a better response from people there.
HCM was the communist leader who started the war; it was a civil war between the communist north (mostly Hanoi) and the democratic south (mostly Saigon).
Despite the tragic loss of 58,000 Americans and over ONE MILLION So Vietnamese, we lost to the north. Shortly thereafter, HCM changed the name of Saigon to HCM City.
People from south VN HATE that name, and never, ever call it HCM City, unless in earshot of Govt personnel.
I go to Saigon about once a year for business. Love the people, hate the pollution.
With 9 million people, it's more populated than NYC.
-von