-Designed for only two horses.

Just done some more specific searches.
In the 19th century newspapers, there are also advertisements for 169 advertisements for 'Barouche Landau' and 183 for 'Landau Barouche'. (The previous total I gave was for all these, plus all mentions of Barouche alone and all mentions of Landau and of Landaulette alone, in advertisements from 1st Jan 1800 to 30th April 1814). As the Morning Post of January 18 1804 describes in its 'Fashionable World' section the Prince of Wales new blue and yellow Landau Barouche, and the next day describes the same vehicle, at the same event as a Barouche Landau, I assume the word order was not important.

Checking the Burney Collection of 17th and 18th century newspapers, there is no mention of either vehicle. There are more than 1300 mentions of 'Landau' in advertisements from 1627 to 1800, but nearly all of these turn out to be references to the place (selling Gazetteers and histories). A search for the word 'Landau' within five words of 'carriage' yields only 30 advertisements, starting from 1760 and all but half a dozen are notices of new taxes in 1785, charging 'coach, chaise, chariot, Landau or other four wheel carriages' the most expensive tax of £3/10 (2 wheeled carriages were charged a quid, and carts ten bob).
Before 1800, there are no references to Barouche at all. There was a place called Baroche,'once the capital of Guzerat' and home to many ruined Hindu Temples and Islamic Mosques in 1783 (now called Vadodara, in Gujerat province) the scene of what seems to be a particularly squalid little war, picked by the East India Company from the sheer greed of some of its directors, the British (or East India Company?)army against the local 'Mahrattoes' when they were not busy fighting each other.(I am guessing this is something to do with the first Anglo-Maratha war, but its hard to tell) The locals were friendly with the French, but independent of them, and from the terms of the peace treaty, and the newspapers discontent with it, it would seem the British suffered a humiliating defeat.
The Italian artist Frederico Barocci (1528-1612)is also referred to as 'Baroche' when some of his painting are exhibited and sold in 1791.
From this I would gather that the Baroche as a carriage is an innovation from about the turn of the century, (so at least a dozen years old by the time we hear of Mr Sucklings). The Landau is older, and the Baroche Landau appeared to be growing more popular every year. (At the 1804 Queens birthday celebrations where the Prince of Wales rode out in his new Baroche Landau, the Duke of Cumberland and Sir Stephen Cotteral had Landaus, the Duke of York a town coach, the Earl of Mansfield and at least half a dozen other notables sported chariots)
Also, there was a tax on the number of carriages kept as well as the type. A single carriage was taxed 8 shillings a year, two carriages were taxed 16 shillings each (ie. £1/12).

References: British Library Burney Collection.
British Library 19th Century Newspaper Collection.