The Oxford English Dictionary argues that its first usage referred to Native Americans. They trace the phrase back to a 1901 account from a man who had spent time among the Sioux some decades prior:

1901 Coll. Minesota Hist. Soc. 9 418 They paid no attention to me, but seated themselves, Indian style, on the ground in a circle in front of me, and beat time by striking on the ground with their sticks. (From the OED)

You can read the full account that this quotation was taken from here.

Some Native Americans consider it mildly offensive " I think this one is a term that needs to be retired, much like sitting “Indian style” (just call it cross-legged!)."
European languages and Japanese use terms such as tailor-style or turkish-style.

The phrase was American English. Lotus position or cross-legged, with three syllables, is used in NZ, Australia and the UK.