Consulting - best though of projects, governance, and on-demand expertise. There are definitely junior resources in play, but the objective is to build on the IP of the consulting firm and to create a external channel for problem solving or delivery that is not dependent on or hampered by internal inertia.
Staff Augmentation - a lot of time this is just billed hourly, and you really are looking at paying a premium for a non-committed spend - if FTEs are "reserved instances" then staff aug contracts are the opposite. You pay a premium for the lack of long-term commitment and to make this look like variable OpEx. my opinion is that this model is over used in most businesses.
Managed Services - longer term commitments to deliver services. Generally the good advice is to contract on "what you want done" rather than the detailed "how you want it done", though that is more philosophical than practical. The expectation is that the service delivery will improve, and that those process improvements, automations, training, and provider IP/Tools will deliver decreasing unit prices for "resource units" over time.
Thinking of consultancies as pure brokers is certainly a mental model you can use, but is not the most nuanced model. I think it tends to be better to acknowledge that there are structural reasons why all of these models are present in organizations and rather to think about how they should be governed.
(disclaimer, my employer is an advisor on how to contract for these kinds of services. My work is not related to contracting, but research about this market)
