1476 times from 1st Jan 1800 to 1st May 1814 in the British Library's 19th century newspaper collection, so its not just Mrs Elton...coachmakers make a distinction between Baroches, Landaus, and Baroche-Landaus. They seem rather heavy (although you can harness two or four horses to them, four seems more normal, and some advertisments for horses boast mention their ability to pull Baroche-Landeaus, as if this was some test of strength)
There are a couple of mentions of Lords purchasing new Baroche-Landeaus (detailed like the dresses of ladies, in the fashion pages) but on the whole, fashionable Lords turn up to the races in other equipage. It seems to me to be more like a Lexus than a Bentley.