Jane could and should have used language which didn’t jump to the conclusion that the word was, in fact, racist.
Walter could and should have reacted less defensively and tried to understand her point of view fully before starting to argue against it.
Both parties are somewhat at fault, but neither error was particularly horrible IMO.
No offense to Walter, but his apparent belief that “Happy as Larry” is a common enough phrase to be a useful comparison is one of the funnier parts of this thread.