Coach, Trainer or Teacher?

A Trainer is a person who educates others on specific topics of specific importance
A Coach is someone who offers advice, ideas and perspective when appropriate and in a way you can use them.

A Teacher is a person who provides formal education for others.

"Teaching is the art of assisting others with discovery." Teachers impart their knowledge and wisdom freely to help others develop and grow within them. Trainers focus on specific areas of development, soft skill and technical, to transfer best practices in action and thought process to others. Coaches observe others to understand their strengths and weaknesses then guide them down paths that will assist in developing or correcting those weaknesses. To be effective, especially in the area of adult learning, you must understand how and when to be all three.
I believe that to be truly effective, you need to be all three and because of this overlapping, there is no difference.

Many ways to define this. Just one option is:
a. teacher tells a group of people about certain knowledge,
b. trainer shares his knowledge while working with a group of people,
c. coach shares his knowledge while working with individual people, but also transfers skills, and provides (mental) support,

What they have in common is transferring knowledge. The difference is important when you have to deliver to a customer.

In most cases a trainer is responsible for delivering information - knowledge passes from the trainer to the trainee, and the trainer's success is measured by the trainee's ability to retain and repeat what they learned.

A teacher is similarly responsible for delivering information, however teachers are often more invested in the development of passion for their subject matter. The measure of a successful teacher is that his/her students are inspired to continue to pursue learning.

An effective coach works with clients to help them find tools and resources to answer their own questions and achieve their own goals. The measure of a successful coach is that their clients become increasingly self sufficient and self directed in the pursuit of their own goals.

A teacher teaches you what is fish(conceptual knowledge).
A trainer trains you how to fish(technical skills).
A coach motivates you to fish(empowerment).