Buddhism broadly speaking offers two kinds of meditation: samatha and vipassanā (Sanskrit śamatha, vipaśyanā). Samatha comes from the root √śam 'to be calm, quiet, to rest'. In samatha meditation we are trying most of all to calm down, and to steady our mind. This in no way involves rumination or dwelling on one's inner world. The archetypal practice is one which involves 'watching' the sensations of breathing, allowing the sensations to fill one's awareness (hence to be mind-full). Note that I do not say "the breath". It is helpful to get away from "the breath" as an entity (what is that in any case?) and to orientate oneself towards the experience of breathing as a dynamic procession of sensations presenting themselves to our conscious awareness. The sensations of breathing offer a good meditation subject because they give feedback on one's state of calm, they change at a pace which does not excite, and they are primarily proprioceptive - i.e. felt as changes in muscle tension in the body - which helps to draw attention away from the primary modes of interacting with the world - sight and hearing. When we allow our minds to be full of these sensations, follow them closely but in a relaxed way, we begin to experience changes in our awareness.
On a good day we find that we are no longer pulled towards other experiences, or towards our own mental chatter. We find that we naturally settle into a relaxed, but focussed state. By attending to experience wisely we can deepen this state until other sensations cease to resister in our mind, and there is only the increasingly subtle experience of breathing. This state can go very deep, and is often described as beautiful, expansive, open, and blissful. One can experience physical rapture, but also other internally generated experiences with a sensory character such as visual imagery. Although we have withdrawn our attention from the world, we find a world within which is at once gloriously alive and yet very refined and subtle. The technical term for this kind of experience is jhāna (Sanskrit dhyāna).
Sometimes Buddhists will frown on talking about meditation experience - straight-forwardly saying that one has experienced jhāna for instance can be seen as "boasting" or "making a claim". This is unfortunate because experiencing a concentrated mind is relatively ordinary, and certainly within reach of anyone who seriously practices meditation in a supportive context. I'm no great meditator and I have had these kinds of experiences. The Buddha's prohibition for the monks is against falsely claiming to be an arahant, and as far as I know there is no traditional prohibition on discussing the experience of various jhānas, nor on claiming to be an arahant if one actually is an arahant. At times a useful discussion is stifled by literalism or over-reacting. I should also say that some Buddhist traditions are distrustful of jhāna. Because it is pleasurable it can become a distraction. I know several people who can easily get into these states, and some of them do say that it can become an end in itself. However my own teachers have always emphasised that jhāna is a means to an end, not the end in itself. Concentrated meditation leaves one feeling calm, happy, and peaceful. Regular meditation encourages psychological integration. The fact of getting concentrated is not in itself very significant or spiritual advanced, but concentration and absorption are useful in preparing the mind for meditation in the second sense.
The essential counterpart to concentrated meditation is vipassanā often translated as 'insight'. The term derives from √paś 'to see' and with the prefix vi- means 'seeing through' - i.e. not insight but through-sight. Using 'insight' as a translation has the unfortunate connotation that we are seeing inside ourselves, suggesting introspection. But what we are doing is seeing through our self not seeing into it. Again this kind of meditation doesn't really involve introspection.
In this style of meditation one reflects on some aspect of experience - the tradition provides a number of templates for this. We might for instance reflect on impermanence, or on suffering. We might reflect on the way things arises in dependence on causes. Other styles of vipassanā practice include visualisations of a Buddha, koan practice, or simply sitting and watching the play of experience. Reflecting this way we aim to see the way experience unfolds, to understand why we feel and think the way we do, not by by dwelling on the content of our own thoughts, but by trying to get underneath this and see how the thoughts that we have depend not so much on the sensations we have, but on the stories we tell ourselves about them. The medium is the message.
This is not like rumination. We don't get hooked on the content of our thoughts, in fact we aim for the precise opposite - to get unhooked from the content of our thoughts. This is why jhāna practice is so useful. With a mind prepared by jhāna meditation we are in a very advantageous position to observe the workings of our mind without being caught up in the content of our thoughts and feelings. Being calm and content we can just be with what we find in our minds. We can also sustain our focus on the subject far more easily.
I don't know much about Zen meditation, or other 'just sitting' or formless practice styles, but as I understand it the formless practices combine samatha and vipassanā aspects. I won't say more, but I do think that formless practice can just about fit into the paradigm I've outlined. And of course meditation is not the only practice. There are also intellectual, ethical, and devotional aspects to Buddhism which are important.
Where a Buddhist can usefully do a little introspection is in the area of ethics. By this I do not mean thinking about morality in the abstract. We cannot really see how Buddhist ethics works by considering hypothetical cases. Buddhist ethics simply asks us to reflect on our own behaviour, and especially our relationships with other people. How do we observe that our behaviour affects those around us? How do we observe it affecting our own minds? We will particularly notice the effects on ourselves in the form of the hindrances to meditation. So if we want to spend time thinking about ethics we can reflect a little on what hindrances to concentration we are currently meeting. Unethical behaviour sets up conflicts and tensions, or scatters our energies which we experience as restlessness, torpor, craving, or aversion. There is often something we can do or cease doing that will be helpful in moving us towards a less conflicted, more alive state of mind. We need not be at the mercy of hindrances.
I hope it's clear that introspection has a role in Buddhism, but that it's role is not predominant, and that in meditation we are not being introspective per se. Of course one will need some self-knowledge, to understand one's own temperament in order to sustain an effective practice. We need to understand our own habitual tendencies in order to effectively counteract them or reinforce them as appropriate. But this knowledge comes as a by-product of attempts to engage with Buddhist practices, and as we interact with other people. The fact that being generous and regulating our behaviour towards others are firmly at the base of Buddhist practice, shows that a lot of self-centred navel gazing is out of place.