Buddha's advice regarding meditation practice

The distinction between Vipassana as a practice to Samatha is a relatively modern one, most likely led by the Vipassana Movement. This is most likely due to the fact that cultivating the Jhana is very time consuming and probably require intensive monastic training and was not possible to be taught in a ten days meditation retreat. So the goal becomes to allow the novice meditators to at least recognize some of the three marks of existence within the practice.

But if you look directly at the Pali discourses — the earliest extant sources for our knowledge of the Buddha's teachings — you'll find that although they do use the word samatha to mean tranquillity, and vipassana to mean clear-seeing, they otherwise confirm none of the received wisdom about these terms. Only rarely do they make use of the word vipassana — a sharp contrast to their frequent use of the word jhana. When they depict the Buddha telling his disciples to go meditate, they never quote him as saying "go do vipassana," but always "go do jhana." And they never equate the word vipassana with any mindfulness techniques. In the few instances where they do mention vipassana, they almost always pair it with samatha — not as two alternative methods, but as two qualities of mind that a person may "gain" or "be endowed with," and that should be developed together. http://www.accesstoinsight.org/lib/authors/thanissaro/onetool.html

This distinction between vipassana and samatha is also not emphasized in Northern Buddhist tradition (i.e. Mahayana Chinese Zen and so forth), but cultivation of Jhana is.

In fact the distinction have led to some criticism:

A group of laypeople who had studied the Abhidhamma together came to Ajaan Fuang to try out his version of mental training, but when he told them to sit, close their eyes, and focus on the breath, they immediately backed off, saying that they didn't want to practice concentration, for fear that they'd get stuck on jhana and end up being reborn in the Brahma worlds. He responded, "What's there to be afraid of? Even non-returners are reborn in the Brahma worlds. At any rate, being reborn in the Brahma worlds is better than being reborn as a dog." http://www.accesstoinsight.org/lib/thai/fuang/itself.html

"But I think he gives different meditation instructions to different people with regard to the path to reach Nibbana."

Yes, according to current difficulties of his audience. For example:

"Develop the meditation of compassion. For when you are developing the meditation of compassion, cruelty will be abandoned.

-- MN 62

"would like to know what kind of meditation practice did the Buddha advise the monks to do, in contrast to lay people."

I've never seeing this distinction being drawn.

"Did he always teach them to practise Samatha first, and then Vipassana?"

Probably not:

Ven. Ananda said: "Friends, whoever — monk or nun — declares the attainment of arahantship in my presence, they all do it by means of one or another of four paths. Which four?

"There is the case where a monk has developed insight preceded by tranquillity. As he develops insight preceded by tranquillity, the path is born. He follows that path, develops it, pursues it. As he follows the path, developing it & pursuing it — his fetters are abandoned, his obsessions destroyed.

"Then there is the case where a monk has developed tranquillity preceded by insight. [...] Then there is the case where a monk has developed tranquillity in tandem with insight [...] Then there is the case where a monk's mind has its restlessness concerning the Dhamma [Comm: the corruptions of insight] well under control. There comes a time when his mind grows steady inwardly, settles down, and becomes unified & concentrated. [...] Whoever — monk or nun — declares the attainment of arahantship in my presence, they all do it by means of one or another of these four paths.

-- AN 4.170

"Did he ever teach anyone to practice the 'dry insight' path, without ever doing Samatha meditation?"

Hard to tell. But it's reasonable to think that he would not teach samatha to an accomplished jhana meditator, and instead, would focus on what he or she is missing to attain nibbāna.

"As a general rule, were there any differences in the meditation practice that he advised between monks and lay people?"

Not that I know of. I didn't really keep track of who the Buddha was talking to, to be able to answer this conclusively. But there were lay disciples accomplished in meditation during the Buddha's time. For example, Citta, a householder, declares his ability to attain jhanas (see SN 41.8). Though I can't tell if he learned it from the Buddha's teachings, at least it seems unlikely any practice was kept away from lay disciples -- besides, lay disciples were known to be reciters of entire nikayas.

"I am aware that the Buddha prescribed basic morality for all, and that some people would reach Nibbana at the same time as hearing him speak about some aspect of Dhamma without doing any kind of formal meditation practice."

It's generally believed the people who did not made much practice under the Buddha's guidance where already very accomplished.

Consider the story of the blind weaver's daughter, Verse 174 of the Dhammapada,

Verse 174: Blind are the people of this world: only a few in this world see clearly (with Insight). Just as only a few birds escape from the net, so also, only a few get to the world of the devas, (and Nibbana).

The girl contemplates death and dying for three years having only heard a single sermon of the Buddha and having learned no samatha technique.

The main points the Buddha stressed on that day may be expressed as follows:

"My life is impermanent; for me, death only is permanent. I must certainly die; my life ends in death. Life is not permanent; death is permanent." (source)

She is ready for stream entry by merely practicing this dry insight. The rest of the story is quite instructive too, but doesn't pertain to dry insight.

As regards the difference between meditation practices of monks and lay people, there are a few traditions today that seem to institute such differences for various reasons - too painful, too difficult, too complicated, however there is no practical barrier to practice per se. Painful ascetic practice of the Dhutanga monks were rarely attempted by lay people for obvious reasons. In the time of the Buddha it wasn't unusual for lay people to take temporary vows of monasticism on retreats lasting even a few years. This was more popular (see caves of lay people where they retreated a few months a year in India) in contemporary religions like Jainism which even included very painful penances.

Meditation is synonymous with Buddhism today, but even in Buddhist countries like Burma meditation almost died out, with most monks eschewing meditation in favor of sutta practice. It was laypeople who sometimes preserved the tradition of meditation.

For example, Sayagyi U Ba Khin (1899-1971) and his disciple S.N. Goenka (1924-2013) are famous lay meditators of Burma who revived and spread a branch of the Vipassana system world wide.

Anapanasati Sutta: Mindfulness of Breathing is the canonical meditation taught by the Buddha. However, the Buddha taught over forty kinds of meditation.