mysql starts and stops servers automatically non stop
Of course you will see your terminal running the grep command .. that's pretty much to be expected. Please learn some console basics (either mac os or free bsd related  as that is the underlaying system that mac os builds on )

If ps -> grep really only shows its own instance then there is nothing else with "mysql" in its process name. 
Which pretty much explains that you cant kill the process afterwards - because the console-command did its work and then the process ended regularly. Hence your second attempt "generated" a new process ID (new process) ...

At the time of your ps -> grep NO "other" mysql process is running.