The terms are used so interchangeably by people that I'm not surprised you might disagree. I looked at a bunch of definitions and tried to distill a useful distinction.
Given that people hire coaches, but usually not mentors, the distinction for me is that a coach is engaged towards a goal and therefore is more directive of what you need to do to get there.
I don't think your example question is a great one for exploring the distinction because it's a single, binary question. But even in your example, despite the coach responding with questions, you describe the coach as pushing you to go do some work: figure out the differences yourself, or come up with a project to explore the question.
In that sense, I see the coach as "setting the agenda" whereas the mentor is having a more open-ended conversation about it.
Hmm maybe you are right and the example is not very good.
Let me try to rephrase it in a way:
I think the main difference for me is that I go to the coach to support me to solve problems/matters by myself and they are there supporting my process but I expect them to have less influence on the actual content/solution itself.
To summarize the coach does not give advices nor they should impose best practices.
While I go to the mentor expecting them to offer me advice and guidance/best practices.
In this I choose (very rarely) a coach to explore problems that I think don't have a universal solution or the solution is subjective like “Should I move to management or continue on the technical path” or “What is best for me: freelancer or employee?”
And I go to mentor to get concrete advice/guidance on specific matters like “How to increase my income as freelancer” or “How to start a new career in X”.
As I write this it seems that for me I see coach as a person that can help me discover the why and the mentor is someone that can help me discover the how.
I've worked with some psychologists giving business consulting, and they would use the definition from the International Coaching Federation (ICF), which is that coaching is "partnering with clients in a thought-provoking and creative process that inspires them to maximize their personal and professional potential".
So in that sense a coach would not set the agenda at all, nor be directive of what you need to do to get there, quite the opposite in fact. They keep asking questions and pushing you to figure out what you need to do to get there, which means that a coach can theoritically help you even if not in the same field as you.
I've also worked in consulting in a past career, and I've had a hired coach, and the line you quote with words like "partnering" I would describe as part of marketing the product. They're not exactly going to say "hire us to push you out of your comfort zone", but that is the role.
You said: "they keep asking questions and pushing" -- that's what I mean by setting the agenda. As I mentor, I don't see my role as "pushing". Questioning, sure. Providing perspective, sharing my stores, yes. Actionable feedback on skills is the closest I'd come to "pushing" and even then, they can take it or leave it.
When I see people talking about coaching, I often see -- directly or indirectly -- some aspect of the role of the coach to be to "bring out their best". I rarely see words like that used to describe mentoring relationships.
You made me think more about this subject with this comment. In a way - due to different incentives - I think you are right about the outcome.
As the coach is mostly hired and the mentor internal it might be that the coach has more incentive to push someone to bring their best while the mentor - having as main focus another job and doing mentorship as a side task - will offer advice/guidance but will not have the same incentive to follow through.
Anyhow I agree there is not a standard definition of what a coach or mentor is and what they should do.
Given that people hire coaches, but usually not mentors, the distinction for me is that a coach is engaged towards a goal and therefore is more directive of what you need to do to get there.
I don't think your example question is a great one for exploring the distinction because it's a single, binary question. But even in your example, despite the coach responding with questions, you describe the coach as pushing you to go do some work: figure out the differences yourself, or come up with a project to explore the question.
In that sense, I see the coach as "setting the agenda" whereas the mentor is having a more open-ended conversation about it.