Need help with page title (Static front page)

The conditional tag is_front_page() is used to specifically check for a static front page, so you should be able to use that to switch between your title formats

<title>
    <?php if(is_front_page()) {
        bloginfo('name');
    } else { 
        wp_title( '|', true, 'right' );
    }; ?>
</title>

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