There is little different between an MSP and internal IT.
They are basically the same thing. In many cases the internal IT is a separate entity that basically bills the company and/or child companies, but is on the payroll of the company.
Yup, the key difference isn't their relationship to the rest of the org, effectively MSP, ITSP, Internal IT, etc. are all external in how they are approached. Only how they are paid really differs and the staff don't always see that.
What makes the two different is that an Internal IT department (even one treated as a consulting group) has only a single top level customer and MSPs have multiple. That's really it.
And that doesn't always make a real difference. If the top level internal IT customer doesn't force all underlying groups to unify under a single IT strategy you get an effective situation of multiple customers, sometimes as you said, even with separate billing.