Question: What is a strong but not too long description of Coaching that you may have used that has worked for you?

Response:

The biggest mistake any coach can make is trying to tell people what coaching is.

So the short answer to your question is: There isn't one.

Even those that exist require explaining to the uninitiated.

My favorite: "Coaching is inspiring an individual or a team to produce a desired result through personalized teaching, expanding awareness, and designing environments."
(looks great on paper and in concept and may be the best definition, but. . .)

So now the long answer.
In the coaching industry we make the fundamental error in believing that coaching is a thing, a product, something we can actually sell. It is not. What people are buying is sense of themselves and who they can be because they are going to be working with you. They are buying the outcome of the relationship. They may be buying relief from a struggle. All of these things are intangible and in the head/heart/soul of the potential client.

The potential client sees/hears/senses something in you that resonates with their needs/wants/sense of who they are and who/where they want to be/go. People ultimately buy based on emotion. As a result it comes back to you. Who are you? How do you make them feel? Are they inspired by you? (Coaching is about inspiring and teaching)

You are your brand! Everything about you from the way you dress to the way you talk to the way you interact with people. This is what they are buying. Not that they want to be you, but rather they want to be they way they feel when they are with you/interacting with you and the sense of possibility or lack there of that you project and are. (to learn more check out day one of my Coaching Business Essentials free eclass)

I'm sure you've heard somewhere that coaching is about BEing not DOing. They are right! Coaching is about who you are. There is a story that at the famous Oracle of Delphi above the entrance was written "First Know Thyself." Great coaches are in total alignment with themselves: what they believe, what they say, what they do. BE-SAY-DO™ is who they are. Since so few people in the world seem to be that way, when potential clients bump up against them something magical happens within them. They become curious, they become inspired.

So my advise:
1. Don't sell coaching. BE yourself
2. Be curious "Be interested rather than interesting" Nobody cares how much you know until they know how much you care
3. Ask permission
4. Let go of the outcome
5. Get your money needs met elsewhere otherwise they will drive your conversations.
6. Become the best person you can be.
7. Develop a spiritual/gratitude practice
8. Practice what you preach/teach/coach.
9. Develop and practice integrity - BE-SAY-DO™
10. After a particularly good round of coaching with someone ask them "So, what is coaching?" or What just happened here.

That's how I got my favorite intro line, At a networking roundtable I stood up when it was my turn and said -"I am in the business of freeing peoples lives" and I sat down. Since I believed it and lived it, it resonated loudly with those ready to hear it. Perhaps its an issue of "When the student is ready the teacher appears." or perhaps its the other way around "When the teacher is ready the students appear" hmm.

Hope this answer helps. I know so many coaches including myself who have gotten trapped in the quandry you posed in your question. Its a Mobius strip.