Very few things, if any, have only one name and names themselves come a variety of types. Sometimes names have overlapping potentially contradictory meanings (I’m looking at you “fruit”). I’d say it’s important to use the right name in the right context for the right purpose but there’s usually a lot of wiggle room in there. The big thing is to avoid confusion.
That said when it comes to proper names of people, the right name is the one they choose to use, period.