A favorable moment or turning point that can be seized to start or change something. Often used with ～を契機に or ～を契機として.
Studying abroad became an opportunity that changed his outlook on life.
この成功を契機として、さらに事業を拡大したい。
I want to use this success as a springboard to expand the business further.
The direct cause or catalyst that sets an event in motion. Often used with ～が契機で or ～が契機となって.
その事件が契機で、法律が改正された。
That incident was the trigger for the law to be revised.
小さなミスが契機となって、大きなトラブルに発展した。
A small mistake became the cause that escalated into a major problem.
I used the move as an opportunity to reconsider my daily habits.
Taking this failure as a turning point, he started studying seriously.
His illness was the trigger for him to quit smoking.
The accident became the trigger for strengthened safety measures.
転機 refers to a turning point in one's life or a situation, whereas 契機 emphasizes the specific event that serves as the catalyst for that turn.