Wordpress - Save values to array upon form submit in admin option page

I have almost done a successful first time plugin, but unfortunately I cannot submit my updated values to the array I've created upon my register_activation_hook.

I need some insights in how Wordpress can handle all the dirty work - or will a $_POST be a better way to do it?

Below you'll see my admin.php which handles all admin related stuff.

<?php 

// Meaning of abbreviations:
// clsc = Custom login shortcode

// Runs when plugin is activated
register_activation_hook( PLUGIN_MAIN_FILE, 'clsc_install');
// Create new database fields
function clsc_install() {
    $clsc_options = array(

        'Login_link'        => '/log-in/',
        'Login_string'      => __('Log in', 'clsc'),
        'Login_class'       => '', // Default is empty to inherit theme styles
        'Logout_link'       => wp_logout_url( home_url()),
        'Logout_string'     => __('Log out', 'clsc'),
        'Logout_class'      => '', // Default is empty to inherit theme styles
        'Account_link'      => '/my-account/',
        'Account_string'    => __('My Account', 'clsc'),
        'Account_class'     => '' // Default is empty to inherit theme styles

    );
    add_option('clsc_options_array', $clsc_options, '', 'yes');
}


// Create admin option page
function add_clsc_option_page() {
    add_options_page(
        'Custom Login',             // The text to be displayed in the title tag
        'Custom Login',             // The text to be used for the menu
        'administrator',            // The capability required to display this menu
        'custom-login-shortcodes',  // The unique slug name to refer to this menu
        'clsc_html_page');          // The function to output the page content
}
/* Call the html code */
add_action('admin_menu', 'add_clsc_option_page');

// Enqueue admin styles and scripts
function clsc_enqueue_scripts() {
    global $wpdb;
    $screen = get_current_screen();

    if ( $screen->id != 'settings_page_custom-login-shortcodes' ) {
        return; // exit if incorrect screen id
    } 

        wp_enqueue_style( 'brokenfruit-shortcodes-styles', plugins_url( 'admin/css/admin_styles.css', dirname(__FILE__) ) );
        wp_enqueue_style( 'bootstrap', plugins_url('admin/css/bootstrap.css', dirname(__FILE__) ) );
        wp_enqueue_script('admin_js_bootstrap_hack', plugins_url('admin/scripts/bootstrap-hack.js', dirname(__FILE__) ) );

}
add_action('admin_enqueue_scripts', 'clsc_enqueue_scripts' );

// Build admin interface
function clsc_html_page(){

    $options = get_option('clsc_options_array');

    ?>
    <form method="post" action="options.php">
    <?php wp_nonce_field('update-options'); ?>
        <div class="bootstrap-wrapper">
            <div class="row">
                <div class="col-md-12">
                    <h1><?php _e('Custom Login Shortcode','clsc'); ?></h1>
                    <p><?php _e('To use for shortcode:','clsc'); ?><br/><span class="shortcode-preview">[custom_login]</span></p>
                </div>
            </div>
            <div class="row" id="login-content">
                <div class="col-md-4">
                    <h5><?php _e('Log in link:','clsc'); ?></h5>
                    <input placeholder="<?php _e('Example: /log-in/', 'clsc') ?>" class="form-control" type="text" value="<?php echo $options['Login_link']; ?>" />
                </div>
                <div class="col-md-4">
                    <h5><?php _e('Log in string:','clsc'); ?></h5>
                    <input placeholder="<?php _e('Example: Log in', 'clsc') ?>" class="form-control" type="text" value="<?php echo $options['Login_string']; ?>" />
                </div>
                <div class="col-md-4">
                    <h5><?php _e('Log in class:','clsc'); ?></h5>
                    <input placeholder="<?php _e('Example: login_style', 'clsc') ?>" class="form-control" type="text" value="<?php echo $options['Login_class']; ?>" />
                </div>
            </div>
            <div class="row top-buffer" id="logout-content">
                <div class="col-md-4">
                    <h5><?php _e('Log out link:','clsc'); ?></h5>
                    <input placeholder="<?php _e('Example: /log-out/', 'clsc') ?>" class="form-control" type="text" value="<?php echo $options['Logout_link']; ?>" />
                </div>
                <div class="col-md-4">
                    <h5><?php _e('Log out string:','clsc'); ?></h5>
                    <input placeholder="<?php _e('Example: Log out', 'clsc') ?>" class="form-control" type="text" value="<?php echo $options['Logout_string']; ?>" />
                </div>
                <div class="col-md-4">
                    <h5><?php _e('Log out class:','clsc'); ?></h5>
                    <input placeholder="<?php _e('Example: logout_style', 'clsc') ?>" class="form-control" type="text" value="<?php echo $options['Logout_class']; ?>" />
                </div>
            </div>
            <div class="row top-buffer" id="account-content">
                <div class="col-md-4">
                    <h5><?php _e('Account link:','clsc'); ?></h5>
                    <input placeholder="<?php _e('Example: /my-account/', 'clsc') ?>" class="form-control" type="text" value="<?php echo $options['Account_link']; ?>" />
                </div>
                <div class="col-md-4">
                    <h5><?php _e('Account string:','clsc'); ?></h5>
                    <input placeholder="<?php _e('Example: My Account', 'clsc') ?>" class="form-control" type="text" value="<?php echo $options['Account_string']; ?>" />
                </div>
                <div class="col-md-4">
                    <h5><?php _e('Account class:','clsc'); ?></h5>
                    <input placeholder="<?php _e('Example: account_style', 'clsc') ?>" class="form-control" type="text" value="<?php echo $options['Account_class']; ?>" />
                </div>
            </div>
            <div class="row top-buffer">
                <div class="col-md-12">
                    <input type="hidden" name="action" value="update" />
                    <input type="hidden" name="page_options" value="clsc_options_array" />
                    <input class="btn btn-primary top-buffer" type="submit" value="<?php _e('Save Changes') ?>" />
                </div>
            </div>
        </div>
    </form>
    <?php
}
?>

Feedback for alternative approaches are more than welcome!

I thank you all in advance and have a great day! :)

via Chebli Mohamed