I am no testing expert but I'll share my opinion anyway
In my experience, there are two types of testing which you are describing (unless I've misunderstood) which is system testing and user or business acceptance testing.
The former tends to be dumb in that they take the system specifications (use cases, perhaps) and ensure that the system behaves exactly as described. They should test as a minumum all of the positive scenarios to prove the system behaves in the way described.Depending on time, they should also test negative scenarios which is to prove all of the permutations that are NOT mentioned in the system spec also behave as described.
For example, when considering customers, there may be many variations which have no impact on the part of the system under test e.g. gender, age, address, length of time as a customer. In theory, you could test different ages, both genders etc but, in practice, the tester should use his/her judgement and test a limited subset.
The user acceptance testing will be more interested in the underlying business processes and what will happen in the real world. In this case, they should test end to end scenarios which should cover the processes you are discussing.
So, in answer to your question, if they are in charge of system testing, they should be able to test exactly what you have specified and prove that this works as defined. If this is not defined clearly enough then you have a challenge which can only be overcome by communication and test script reviews (which is what you appear to be doing).
If they are in charge of user acceptance testing then they should have experience of the business and either team members have direct knowledge/experience of the business or have access to business people.
I have recently been in your position regarding system testing of a complex system with many business rules and a complicated underlying framework. The only solution we found was frequent communication and making yourself available to review test scripts. provide clarification and, ultmately, undertaking some focussed testing to improve confidence levels.
Sorry for the rambling reply - I hope this was helpful