Usage of theme name in functions

In short: it’s the so called »textdomain«.

The longer explanation:
If you look accurate on your first example, you’ll see, that this parameter is not the second parameter for the_content but for the Funktion __():

<?php 
the_content( 
    __( 
        'Continue reading <span class="meta-nav">&rarr;</span>', 
        'twentythirteen' 
    ) 
); 
?>

This function is part of the internationalisation (»I18n«) functions to make text translatable.

The textdomain acts as an identifier for translation files, which are typically stored in wp-content/languages for themes. So if you want to use this theme in German language for example, the language files would be named like twentythirteen-de_DE.mo and twentythirteen-de_DE.po.

de_DE is the so called »locale«, which is typically defined in the constant WPLANG which is (since WordPress version 4.0) stored in the option WPLANG.

.mo files are binary files, used by WordPress for the translation and .po files are editable (text) files to define translation for each string used in the source.

Some useful sources on internationalisation is the official documentation:
http://codex.wordpress.org/I18n_for_WordPress_Developers