Is there any way to open an "incognito" Terminal/shell?

Solution 1:

You did not specify which shell you are using but:

The HISTFILE environment variable defines where the history is stored. With

unset HISTFILE

you can tell bash to stop recording commands. You can re-enable it with

export HISTFILE=${HOME}/.bash_history

From man bash:

HISTFILE
       The name of the file in which command history is saved (see  HISTORY  below).   The  default
       value  is  ~/.bash_history.   If unset, the command history is not saved when an interactive
       shell exits.