Staying within the 20 year rule: in 1995, twenty years after the end of the VN war, did some officials still use the old name (Saigon)? How about "the people"?
My family fled Saigon in 1975. I never got a chance to ask my grand parents what they would have called the city now. My mom still says Saigon.
After the war, Saigon was merged with Gia Định province. It was renamed to Ho Chi Minh city in 1976. So technically, Saigon and HCM city are not the same.
In formal writing like commercial contracts or legal documents, "Ho Chi Minh city" must be used since it is the official name. In daily speech, whichever is fine. I live in the north and people say Sài Gòn all the times, keep it short and sweet.