Very difficult to qualify what is and isn't a Kidney Bean, perhaps there is more than one definition... in the past, that was a catch-all term for
P. vulgaris, at least in British English. It appears that no distinction was made for of seed shape, climbing habit or whether the variety was grown for green pods or dried beans.
The English 3rd edition of Vilmorin's
The Vegetable Garden, 1920, has a chapter titled 'Kidney Bean, or French Bean', reserving the title of 'Common Bean' for
Vicia faba. Edible-podded varieties with round seeds are even referred to as Kidney Beans, only to then be described as having 'egg-shaped seeds'. It's apparent that by this point the term was thoroughly detached from a prerequisite for, or connection to, kidney shape.
White Coco, or Lazy Wife, Kidney Bean. Stem green, about 6 1/2 ft. high; leaves of medium size, stiff, rather long and pointed, of a dark, rather dull, green, and slightly crimped; flowers white; pods of medium length, rather broad, green, each containing five or six white egg-shaped seeds, about 1/2 in. long, nearly 1/2 in. broad, and over 1/4 in. thick.
But it appears that any obligation for true kidney-like appearance was disregarded even in the early days. Gerard cites Kidney Beans in 1597. Here again this designation is used broadly for
P. vulgaris; he mostly describes climbing sorts and distinguishes them by the colour of the seedcoat. They're eaten exclusively in the immature pod stage, boiled with butter. The illustrations include some bean seeds that aren't particularly kidney-shaped.
When the term evolved from denoting 'all vulgaris types' to just 'kidney-shaped beans with red skins' is unclear, but must have been some time in the latter half of the 20th century, as a Carter's catalogue I have here from 1934 is still calling all dwarf varieties Kidney Beans.