If you can't tell the difference between using a name and using an exaggerated skin color then I don't know what to tell you.
So,if Miami called their teams "The Miami Natives" would that be O.K. ?
Maybe,when they were considering the name change,Miami should have found out what
the descendents of the Miami's, who are now the "Miami Tribe of Oklahoma" thought about the team name.
As I understand it and remember it, Miami had the longstanding support of the Miami Tribe. They even had a representative of the tribe who would do a genuine war dance in native garb at some of their games. I remember seeing this on more that one occasion. However, at one point the tribe withdrew their support, and that's when they made the name change. So on that level, Miami's decision seems reasonable to me.
As a side note, it's interesting the term "redskin" is actually an English translation of an Indian word (forget which tribal language). This traces way back to some of the first meetings of the aboriginal peoples and the Europeans. The tribe in question called the Europeans a word translated into English as "paleface" and then they coined the expression translated as "redskin" to refer to themselves. Let me say that I haven't researched this etymology myself, but was told this by an expert in the field. If true, it's quite interesting and puts, perhaps, a slightly different context for these discussions.