When should the second drupal_bootstrap() parameter be used?
Without looking more closely, I can't say why statistics_exit() doesn't use the second parameter, but to answer your question...

The drupal_bootstrap() function only ever needs to be called with $new_phase = FALSE to prevent an infinite loop inside the core bootstrap code. Unless you're modifying core, you'll always want to leave the second parameter at its default value (TRUE).

The key is in this part of drupal_bootstrap():

// When not recursing, store the phase name so it's not forgotten while
// recursing.
if ($new_phase) {
  $final_phase = $phase;
}


Calls to drupal_bootstrap() outside the bootstrap process always specify the final phase to bootstrap to (or only get the current phase, in which case, the second parameter doesn't matter). When you call drupal_boostrap(), you'll be initiating the bootstrap process and setting the static $final_phase variable, so you'll need $new_phase = TRUE.