bash の最初の引数は、引数が渡されたり渡されないときに異なります。

bashマニュアルページによると：

   -c        If the -c option is present, then commands are read from  the
             first non-option argument command_string.  If there are argu‐
             ments after the command_string,  they  are  assigned  to  the
             positional parameters, starting with $0.

もっと下へ：

   If arguments remain after option processing, and neither the -c nor the
   -s option has been supplied, the first argument is assumed  to  be  the
   name  of  a file containing shell commands.  If bash is invoked in this
   fashion, $0 is set to the name of the file, and the positional  parame‐
   ters  are set to the remaining arguments.

つまり、$0bashをどのように呼び出すかによって動作が異なります。