Does Zen distinguish between concentration and insight meditation?

David, in answer to your question, you are not supposed to be examining your mind or anything else in Zen meditation. The practice is very simple, yet difficult. The focus is on the breath, and the posture is important. When thoughts arise, as they will on their own accord, the attention is simply brought back to the breath. They are not labeled or judged, they are simply "thoughts". Sometimes we get caught in them in the meditation, and when we notice this, we gently return our focus to the breath.

The first question that naturally comes to mind is, how on earth can something that simple wake people up to enlightenment. Well, it can. You are doing it by not doing it. If you attempt to get any insight or anything else from the practice, it will not work. But if we let go of that concept, or any and all ideas and concepts, then it works. It is a lifetime thing too. Over time, one's understanding of the mind increases. As Dogen said, Buddhism is about understanding ourselves (our true self, not the ego self). If we understand ourselves, then we understand everything. But if we do not understand ourselves, then we do not understand anything. Enlightenment is experience (not simply insight) to the entire universe, and the experience that all is one, and all is empty, clear, impermanent and in constant change. Yet there is something that is not impermanent and is not change. Zen and some other spiritual practices understand that words will not get you there, it is beyond language, language is a bit of the problem, It is the experience that is the truth.

In Rinzai, we don't necessarily call what we do insight meditation, but there are some similarities. A koan is supposed to provide "insight" into existence. They all relate an essential quality of mind (big "M"!) that is otherwise obscured or veiled in normal consciousness. Like you mentioned, arriving at those insights requires that we first develop a certain level of samadhi. Once that is established and stabilized, we begin to look for a koan's answer in mushin - or no mind - the blankness that appears when concentration is established to a certain degree.

What is interesting is that one probes their koan in meditation, one's subconscious obstacles and barriers begin to become more evident. This is an important part of practice. I'd go so far as to say that it's even more important than answering your koan. Some of the barriers one encounters are related to the koan, but others are native to our own karma, ego, hang-ups, habits, mental formation, etc. These all have to be acknowledged and worked through as one grows in their practice. From a practical perspective, I suppose this is the aspect of Rinzai training that is most similar to vipassana. I think it's important to distinguish, however, that in Zen, when those obstacles begin to appear, we don't [usually] explore them according to the three marks of existence as you might in insight meditation.

Personally, I'd love to here a Soto take on this. My cursory exposure to that tradition indicates to me that they might have a more shamatha approach to their sitting.

I'm far from an expert and even more far from knowing all the technical words, but I sit in Soto tradition.

As @enenalan already mentioned, Soto has a more shamatha approach, called shinkantaza, or just sitting. (@enealan: there is your answer, even if I just acknowledged to not answer another answer ;-) )

The meditation in Soto, at least in the tradition of my lineage, is an aimless meditation. It is not only aimless in terms of no insight, no solving koans, no tinkering with truth and such stuff, but also in terms of not meditating for any benefit. We are not supposed to sit to become better people, get insight, be calmer or whatsoever.

While you are in (Soto-)Meditation, you are supposed to watch or notice [was: examine, see comment] our mind. Therefore there is a component of insight. But we should do so as a spectator, not as a judge. The insight comes naturally, over time, and is a side-effect and not thought to be the goal of the practice.

On the other hand, to stay focused, not to let your thoughts wandering, not to start daydreaming or become sleepy. To watch and not not judge our thoughts, how they emerge and vanish, requires a huge amount of concentration.

As a Conclusion: Soto knows (as far as I am Soto) only on kind of meditation, and that covers the concentration aspect. Insight grows over time naturally and is not active fostered during meditation.