How does BER performance depend on gray or natural binary?
You are confusing symbol errors and bit errors. What you say is true for the symbol error rate; it only depends on the signal-to-noise ratio (SNR). However, depending on the coding, i.e. the mapping of the bits to the symbols, one symbol error can cause one or more bit errors. The most likely symbol errors are the ones where a symbol is confused with one of its closest neighbors. Gray coding makes sure that those most likely symbol errors cause only one bit error. Other types of coding will result in more bit errors per symbol error. This is why Gray coding minimizes the bit error rate for a given symbol error rate (i.e., a given SNR), if one assumes that symbols are only confused with their closest neighbors, which is very likely for moderate to high SNRs.