I would guess some kitchens in the US have come to use these terms more loosely. Traditionally the word stage comes from the French “stagaire” which means apprentice or intern. Many famous chefs got their start begging to work for free in well known kitchens to gain experience or a future position.

A trail is more of a brief working audition. The purpose is to showcase your skills and ability to perform in the kitchen where you are seeking employment.