Translation guide
The sport of badminton is referred to by the loanword バドミントン. This is the standard term used in all contexts.
Referring to the game of badminton
The standard Japanese word for badminton, borrowed from English. Used in all contexts.
週末に友達とバドミントンをした。
I played badminton with my friends on the weekend.
バドミントン部に入りたい。
I want to join the badminton club.