Yes, it is almost complete, but you still need the cookie in your definition.
<IfModule LiteSpeed>
CacheLookup public on
RewriteRule .* - [E=cache-control:max-age=3600]
RewriteCond %{HTTP_COOKIE} Cookie_Name_1 [NC]
RewriteRule .* - [E=Cache-Control:vary=is_logged_in_cookie]
RewriteCond...