...although Saigon is still a district of HCM..
Not that it's a big deal, but I think this is a bit off. I'm not Vietnamese, so if you are, maybe I'm obviously wrong and you are right. But I've lived here a long time now. There is no Saigon District. There are many numbered districts, and several with names like Binh Thanh and Thu Duc. Locals who were here when it was a much smaller city will still refer to the central-business area, District 1, as Saigon. But they don't ever use it with "District" in the name, like Quan Saigon. Most everyone else, unless from Hanoi specifically, calls the entire city Saigon in verbal exchanges or uses the district name/number for specifics, even most everyone from the north who now lives here. In written things, HCMC "TPHCM in Vietnamese" is still used and especially on anything government related.
As for the conflict, Saigon was Saigon for a few hundred years before the conflict. But of course it is right after the fighting they changed the name, as one does when taking over a city.
Apparently few people refer to it as Ho Chi Minh City, but still call it Saigon and the airport code is still SGN
Some times these change, sometimes they don't. BOM is still used for Mumbai. Also they can change with new airports. For a long time there was an airport in the USA with a code, and then a new (or maybe just enlarged?) airport in Europe took over that code that was previously in the USA. While of course the US and European airlines new this, an airline in the alliances from another continent changed their use of it in reservations, but not in their award chart. People on travel message boards like this kept it pretty secret, but you could fly for a pittance to Europe in terms of using air miles. I never did, but it was popular for a year or two until someone let the cat out of the bag and that airline updated their IT systems.