I'm not quite sure I understand what you're trying to do, but if you're trying to load "First Header" for your home page and "Second Header" for a different home page, some if/else logic should sort that out for you.
I do something like the following if I want a huge hero header on the home page, and something more subtle elsewhere.
In your main header file that loads on every page, where you want to swap headers, do something like this:
<!--The usual head stuff goes here-->
<?php wp_head(); ?>
...and so on...