They don't. But, there is a small connection. The shuttle cock in badminton is sometimes referred to as a birdie. And a birdie is a score of one under par on a given hole in golf.
Serving in badminton is opposite from the form used in tennis. To serve, hold the bird in one hand and hold the racket below it. Then drop the birdie and hit the center of the rubber tip with an underhand motion.
The goal is to hit the shuttle, or the 'birdie,' over the net and land on other side of the net. If a person hits it into the net or off the court they will lose that rally. You can hit the shuttle only once before it goes over the net. When serving, one cannot serve like in tennis, it must be below your waist when you hit the shuttle.
One under par
An eagle is two strokes under par, a birdie is one stroke under. An eagle is a better score.
A score of one under par
A birdie is one stroke under par on a given hole.
The answer is birdie. It is one under.
A birdie is a hole played one stroke under par.
A "birdie" is not a specific score, it's one under whatever par for the hole is. If it's a par 3 hole, then a birdie is 2; if it's a par 5 hole, a birdie is 4.
Under par. Birdie, eagle or albatross.