It sounds to me like the first technique you mentioned was the do-nothing technique; that would not be mindfulness meditation then. Mindfulness meditation often involves the focussing of attention on things happening in ones present experience, mostly being the breath, bodily sensations and sounds, but can also involve paying attention to ones emotions and where they lay in the body and mind, as well as ones own thoughts. The only time I know of where you wouldn't be using concentration on one specific aspect of experience in mindfulness meditation would be when you simply be aware of all those things I mentioned at once, but you are still manipulating where your attention is being put, so if there's no amount of concentration being used, then it's not mindfulness meditation.