Christian Dior was re-branded to just Dior happened during Galliano's time at Dior. It wasn't immediate. Dior Joaillerie and Dior Homme were also born at that time, which did away with the "Christian" part of the name. In the years before, the Christian Dior label may have been tarnished with licensing agreements where the label could be found on mass-market lingerie and men's dress shirts and belts sold at stores like Sears. Bernard Arnault ended those licensing agreements (took him years to buy some of them back).
Being simply "Dior" gave the brand a fresh identity. Older boutiques were renovated, and included the name change to just Dior. I remember that around 2004, the dust bags and boxes were re-branded to Dior as well for consistent brand messaging.
This is what the logo on the boutiques used to look like, even during the beginning years of Galliano's time at Dior:
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You can see with the Galliano-era ad that advertising has already moved to just "Dior" and looks a lot younger, but when juxtaposed with the older boutique style (in this photo showing a construction facade next to the boutique entrance), it looks like two separate brand identities.
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The ads back then featured the full logo as well. This logo does not match the Galliano-era ads.
Changing Christian Dior to simply Dior not only gave the brand a fresh image, but also made it look younger.
Dior still maintains the full "Christian Dior" label in all of its handbags and women's RTW. The shoes have also featured the full Christian Dior logo since 2007. The shoes simply said "Dior" for a few years before 2007.