Infinite loop when logging out using custom login form

I am speculating here after reviewing your logic/comments above, yet let’s try as follows

Possible issue:

During login, although you are setting the auth cookie, you are not setting the current user and this is potentially the cause of the break-down upon logout due to is_user_logged_in returning a false-positive/negative result.

Possible resolution:

In your validation-room.php add a call to wp_set_current_user, as when entering your callback mb_page_redirects on the template_redirect hook, you are relying on is_user_logged_in to function correctly, which itself checks the state of the global $current_user variable.

Under the hood is_user_logged_in calls wp_get_current_user which calls _wp_get_current_user which fires a filter determine_current_user that calls two internal callbacks wp_validate_auth_cookie and wp_validate_logged_in_cookie.

It is possible that the initial improper validation and later subsequent calls to wp_logout and wp_set_current_user are somehow causing an issue.

$mb_field_room  = ( isset( $_POST[ 'mb_field_login' ] ) ? $_POST['mb_field_login'] : null );

if( $mb_field_room || wp_verify_nonce( $mb_field_room, 'mb_action_login' ) || !empty($mb_field_room) ) {
    $mb_meeting_log = isset($_POST['log'])  ? $_POST['log'] : null;
    $mb_meeting_pwd = isset($_POST['pwd']) && $_POST['pwd'] == '{mb_login_password}' ? 'password' : null;

    $user = get_user_by('login', $mb_meeting_log);

    wp_clear_auth_cookie(); // let's be sure about it...
    wp_set_auth_cookie( $user->ID, true, true );
    wp_set_current_user( $user->ID, $user->user_login );
}