It goes back even further than that.

An 1865 dictionary of American English uses “boot” instead of “trunk” to refer to the… well trunks that were strapped to the front and back of a coach. (A coach being a specific kind of horse-drawn carriage, which takes its name from the village of Kocs in Hungary where they were popular.)

https://archive.org/details/americandictiona00websuoft/page/152/mode/2up

https://www.etymonline.com/word/coach

In that 1865 dictionary, a Dickey (or Dicky) is defined as “A seat behind a carriage, for servants &c”, and a Rumble as “A boot with a seat above it for servants, behind a carriage.”

https://archive.org/details/americandictiona00websuoft/page/1156/mode/2up

So, originally in American English, the trunks strapped to the outside of a carriage were called “boots”, and the seats above them were “rumbles”, and maybe when there was no “boot”, just a seat for servants they were called “dickies”.

In Indian English somehow the “seat on the outside of a carriage” became the “compartment in the back of a vehicle for storing things”. In British English they kept the name “boot” when it changed from an external box to a box that was part of the vehicle itself. And, in American English, they switched to calling it a “trunk”, most likely before it actually became part of the vehicle.