You were close with your second one-liner. From perldoc perlrun you'll notice that -p wraps your code in:

LINE:
  while (<>) {
    ...             # your program goes here
  } continue {
    print or die "-p destination: $!\n";
  }

which prints everyline by default, while -n wraps your code in:

LINE:
  while (<>) {
    ...             # your program goes here
  }

Note, there is no automatic printing here. So if you use -n instead of -p in your second one-liner, you would have what you want.