Trying to remove duplicate jquery scripts

When you load jquery from WordPress includes, you will need to use jQuery instead of the dollar sign $, as the latter is reserved for other libraries.

jQuery( document ).ready(function( $ ) {
    // Submitform and colorbox init, both will work here.
});

But also make sure to avoid the script tags as much as possible, but with wp_enqueue_script in the functions.php (all within <?php ?> )

function add_our_theme_scripts() {
   $theme_url = get_bloginfo('template_url') . '/js/';
   // Add the wordpress version of jQuery. 
   wp_enqueue_script( 'jquery' );
   wp_enqueue_script( 'jquery-submit-form', $theme_url . 'submit-form.js';
   // similarly for colorbox if it is added through the theme.
} // END function add_our_theme_scripts()

// It should be hooked to init or wp_print_scripts and not wp_head.
add_action( 'init', 'add_our_theme_scripts' );

Now if you wish to make it more simple and manageable put the previous code in a separate file and add that to the above function too.

jQuery conflict

You can also try jQuery no conflict mode like below.

  // example var myJQ. 
  myJQ = jQuery.noConflict();

  myJQ( document ).ready(function( $ ) {
      // scripts here.
  });

But be careful though, $ sign will not be available after you enter noConflict mode, so make sure you use myJQ ( example here ) consistently.