There's a lot of different sorts of testing. Many of them overlap. Many use the same tools. Many are specializations of other more general terms. Often they blur together. People argue about the terminology all the time.
You're correct that they all have the end user in mind, but they are different.
This is a specific form of an acceptance test where a subject-matter expert, ideally the client or their representative, tests the software. This is in addition to functional and acceptance testing done by QA. It's designed to simulate, as closely as possible, an actual end-user using the software; the tester is asked to perform a bunch of common tasks with the new system, but not given specific instructions nor coaching on how to do it.
For example, if you were creating a site for an airline, they might be asked to register, login, book a flight, make a payment, check in, check their flight status, and so on.
This is blackbox testing done by the QA role. It verifies the thing does what it's supposed to do; you give it inputs, you check the outputs. Typically this is testing against the specification and/or requirements document.
"Functional" here doesn't refer to code functions, but that the system functions as expected. Testing specific functions is unit testing.
They can be purely functional, "when I do X I get Y". They can be about resource use, "when I do X it uses no more than Y memory/time". Or about error checking, "when I give it garbage I get a well formed error". Anything that validates it meets the requirements.
Test Case Scenario
Sounds like Scenario Testing: this uses stories, similar to user stories, that help a tester work through a complicated testing scenario. Scenario testing tests complicated combinations of things which might arise during actual use and often cut across multiple systems.
An example of a test scenario might be: "in the middle of processing the system runs out of disk space; verify an admin is notified, that processing resumes once space is cleared, and that no data is lost".
A User Acceptance Test might use Scenario Testing.
These are my rules of thumb:
- Unit testing: does this one function work?
- Integration testing: do the functions work together?
- Functional testing: does it function as required?
- Acceptance testing: is it acceptable to the client?
- Regression testing: does it still work like it used to?