For all three services (coaching, consulting, training), there's two sides involved: Service provider and service taker (client). Both make assumptions about themselves and about the corresponding other party. For simplification purposes, we will not list all assumptions, but focus on the essentials that are related to learning.
Interpretation of the modelThe first thing we should be clear about is that these are all just assumptions.
Since they are assumptions, it's good to clarify that both client and service provider have the same understanding of these assumptions beforehand, since they define expectations.
These assumptions are not axioms, since each variable can be verified objectively by asking questions and observation. Neither client nor service provider should turn any of these assumptions into dogma and insist they be true regardless of reality. You must accept that any of them may turn out invalid at any time.
For each of the three roles, the service provider is expected to have a clearer understanding of the big picture than the client. As such, regardless of whether you are coach, trainer or consultant - you need to be actively on the lookout whether the above assumptions are valid. One skill you need to bring to the table is the ability to realize when they are invalid, because it breaks the model of your role. When they are invalid, you must take steps beyond dogmatic insistence on the definition of your role in order to move into the direction of success.
The main reason for getting help is that you don't really know what you may need to know. Your initial assumptions may be invalid because of what you did not know. Given better information, you need to adjust your course of action accordingly.
Roles are really just transient.
It's probably easiest for the trainer who provides a specific training service to stick to the agenda and simply leave. Worst case, the training did not help and the very limited few days of training are wasted. However, even trainers often add coaching techniques and modules to their trainings where they actively generate learning with their clients. In rare cases, that may turn into consulting sessions. When a trainer leaves, the client should have an appetite to try out the training knowledge and learn more.
The line is significantly more blurry for coaches and consultants.
The best thing a consultant can do is enable the client to solve the specific problem and related problems individually by producing learning within the organization. This may include domain-specific trainings in skills the consultant provides and coaching key players in doing the right thing. When a consultant walks out, the client should be able to say "From here, we can move forward by ourselves." - which is the best thing a coach would also hope to achieve.
For a coach, the main difference to a consultant is that there is no specifically defined problem initially and that the coach is not expected to come up with a solution. However, a good coach should understand that there are situations where simply giving a directed pointer to existing solution in order to instill an appetite for learning and experimentation is a good way forward. That's a training situation. Also, sometimes, the coach needs to take carefully considered shortcuts in the learning process to prevent irrecoverable damage: That's consulting. Depending on where on the learning curve the team is, that can be quite a big part of the job.
Assume that coaching and consulting are two distinct roles and that you are either-or is a false dichotomy. In the same breath, it's an even worse misinterpretation to consider one of them"superior" to the other, because both simply rely on different assumptions. A good consultant will use significant coaching techniques in context, and a good coach will use significant consultative techniques in context as well. "Context" depends on observation and interpretation and is usually very mixed. Be ready to accept this mix. Your actions then also need to reflect this mix.
Being dogmatic on one specific role and insisting on the above assumptions as axioms is done only by people who are unable or unwilling to consider the systemic implications of their own actions. That's snake oil. Caveat emptor.