when people type in http://site.com/activity, did you want them to be redirected somewhere else?

no

when people type mydomain/elggdirectory then redirect to login if they are already not logged in.

because anybody can see all activity by typing http://myelggsite/installation-directory

You mentioned that you are using the Private River plugin. Try changing the code of the private_river_page_handler() function in this plugin's start.php to

function private_river_page_handler($page) {

if (!elgg_is_logged_in()) {
    forward(REFERER);

    return false;
}

    $page_type = elgg_extract(0, $page, false);
    if (!$page_type) {
        forward('activity/owner');
    }
    if ($page_type == 'owner') {
        $page_type = 'mine';
    }
    set_input('page_type', $page_type);

    // content filter code here
    $entity_type = '';
    $entity_subtype = '';

    global $CONFIG;
    require_once("{$CONFIG->path}pages/river.php");
}

The modification is highlighted in bold.