You also have to change all occurrences of Bartik in the files to your new name.
I don't think there are any such occurrences. Which "files" do you mean?
go to drupal core themes/bartik/templates. Not /sites/all/themes. Copy the page.tpl.php and node.tpl.php to your /bartic-copy/sites/all/themes/ Copy the /themes/bartik/template.php to your /bartic-copy/sites/all/themes/
Then open template.php in your subtheme folder and change every instance of bartik_preprocess to your subthemename_preprocess.
page.tpl.php is where drupal calls the theme bits and piecs from to fashion your page. You can manipulate this as you want but I suggest you don't.
You can copy the /bartik/css folder over to your subtheme /sites/all/themes/yoursubtheme/ so you can override core theming using the correct terminology.
Now the trap you will run into is: When you update Drupal core, it updates the core bartik theme which calls on a lot of core includes and this can break your bartiksubtheme. Been there, done that.
Hope this helps Roger