How to print different informations for different post types inside The Loop?

Of course, you can use conditions like this inside the loop.

if ( 'event' == get_post_type( get_the_ID() ) ) {
  // do something
}

Full code would be

// The Loop
    
    if ( $query->have_posts() ) : ?>

    <?php if ( is_home() && ! is_front_page() ) : ?>
        <header>
            <h1 class="page-title screen-reader-text"><?php single_post_title(); ?></h1>
        </header>
    <?php endif; ?>

    <?php
    // Start the loop.
    while ( $query->have_posts() ) :
        $query->the_post();

        /*
         * Instead of publishing all the content, on home page we only get the title, the thumbnail and an excerpt
         */
    the_title( '<h1 class="entry-title">', '</h1>' );
    the_excerpt() ;
    
    // Test if this is an event and then print details
    if (is_singular('event') ) {
        eo_get_template_part( 'event-meta', 'event-single' );
    }
    if ( 'event' == get_post_type( get_the_ID() ) ) {
      // do something
    }

        // End the loop.
    endwhile;