There was no unitary term, and few Early Modern Southeast Asians believed in a coherent geographical unit called Southeast Asia.

The closest thing you had was "Below the Winds," originally a Persian term used for the entirety of Southeast Asia in reference to the fact that Persian merchants in the Indian Ocean reached Southeast Asia by following, or descending down, the monsoon winds. Though originally a term no less foreign than "Indochina" or "East Indies," the word soon gained currency throughout Islamic Southeast Asia through the Malay translation bawah angin. And, by analogy, all areas not part of Southeast Asia became the lands "Above the Winds" (atas angin).

The earliest evidence of local usage comes from August 1599, in a Malay-language letter sent to the Dutch East India Company by Sultan Said of Ternate, who was at the time the most powerful ruler in the easternmost Indonesian region of Maluku and had a near-monopoly on clove spice production:

[This letter is] from the land below the wind [dari tanah di-bawah angin]... [from] the king of Maluku, the possessor of the clove trees so famous in the land above the wind [telah sampailah kepada raja Maluku yang empunya pohon cengkeh yang terlalu masyhur di tanah atas angin].

In 1602, the sultan of Aceh, this time in the westernmost part of Indonesia, describes himself in a Malay-language trading permit to English merchants as:

The lord in power here below the winds, who holds the throne of Aceh and Samudra and all the countries adjacent.

So by c. 1600, the term bawah angin was fairly widespread in Malay-speaking Islamic Southeast Asia.

The word was slower to catch on in those parts of Islamic Southeast Asia where Malay was not the prestige language. But in the 1660s, the sultan of Banten, whose court spoke Javanese and Sundanese, referred to his subjects in an Arabic-language letter as "people below the winds" (ahl taht al-rih). By the early nineteenth century, the Javanese-language autobiography of the Javanese rebel prince Pangeran Dipanagara referred to the Ottoman sultan as "General from Above the Winds" (Jéndral diatas angin).

Similarly, in South Sulawesi, where the local languages Makassarese and Bugis were the prestige languages and not Malay, seventeenth- and eighteenth-century sources freely use the term "Below the Winds" to mean "(a place in) Southeast Asia," "Above the Winds" to mean "a place not in Southeast Asia," and "Above and Below the Winds" to mean "the entire world":

[The Dutch East India Company told Arung Palakka, late seventeenth-century king of Boné, that] you have a father and a mother in the Company. The Company and Boné are like the black and the white of the eyes which cannot be separated. If your mother and father, the Company, are poor, you too will be poor. Live below the winds, in your inheritance in Boné and Soppéng. You and your brother will have power, and you will rule in the lands below the winds.

--

[The king of Soppéng told Arung Palakka to be careful when making deals with the Dutch in their colonial capital of Batavia so that] the great men who rule there do not turn the treaty around because I have always said that the people above the winds are shrewd and intelligent.

--

Following the words a u ta u a pa la la ta ra ja sa ha ma ra da [meaning unknown] makes very great the prosperity of all the communities Above the Winds and Below the Winds.

I'm fairly sure some Indonesian sources refer to the Mainland kingdoms like Burma and Thailand as "Below the Winds," though I can't recall any off the top of my head. Persians and Arabs certainly did call both Mainland kingdoms Below the Winds, and in the eighteenth and nineteenth centuries, ethnically Burmese Muslims actually referred to themselves as Zerabadi (zirbadat is "Below the Winds" in Persian).

The non-Islamic areas of Southeast Asia didn't have similar concepts. In Early Modern Burma, for example, the world was divisible into "one hundred and one races," themselves sorted into five broad categories: the Myanma, the Tayok, the Shan, the Mon, and the Kala.

• The Myanma were the Burmese and Arakanese.
• The Tayok were the Chinese.
• The Shan included most Tai-Kadai-speaking peoples, including the actual Shan ethnic group and the Thai and Lao, as well as the Kachins and Khmers.
• The Mon included the Mon people of Lower Burma.
• The Kala were everyone else, including all Indians, all Middle Easterners, all Europeans, and everyone from Archipelagic Southeast Asia.

From the Burmese perspective, the Burmese and the Thai were peoples as different as the British and the Chinese, and the British and the Malays were more similar peoples (both being varieties of kala peoples) than were Malays and Burmese.