There is admittedly a difference between staff augmentation and McKinsey style strategic “consulting”. The distinction is usually who owns the project?
If the client company owns the project and you are just coming in as a warm body, that’s staff augmentation.
But if the client company is putting out Requests for Comments to different companies and they sign a Statement of Work and your consulting company comes in and does the work, that’s “consulting”. In the latter case, you don’t usually get let go as soon as there is no work for you - ie when you are “on the bench”.
Even if you are a more junior employee at the latter company where you are more hands on keyboard than flying out to meet customers and sometimes you might even be doing staff augmentation for the client, it still feels differently.
My consulting company has internal employee events, is responsible for my pay, performance, etc - not the client.