I am coming from the interviewer-led McKinsey style case and I can definitely tell you that it makes a difference for the candidates I coached in how they study and how they approach cases.
A one size fits all approach does not work in my opinion, especially when we are talking about the level of competitiveness and the low offer rate in MBB. You want to be well prepared for each type of case and use the different logistics at play for your advantage.
That being said, the same principles of problem-solving prowess apply in both types of interviews. However, interviewer-led cases are very formulaic in nature and there is a certain checklist of habits (what do and when) you should employ to go through in order to maximize your performance (more on that below). And this checklist is quite different from an interviewee-led case.
While in an interviewee-led case, the main goal is to reach a sound recommendation after going through your issue, analyzing data, etc., the main goal in a McKinsey case is to provide sound and self-standing answers to each individual question (structure, math, exhibit). Think of the latter as a series of mini-cases. More often than not, there will not even be a synthesis/summary in the end (unless your structuring part was weak).
Hence, you should tackle each individual question in a specific way. When you know what actions you should do for each type of question, it is easier for you to focus, be creative, and be structured throughout each question. I think they are much easier to solve as you have to worry only about one problem at a time.
As a result, in a McK case it is also much less about the correct result or solution (except for the numerical part), but much more about how you approach, how you solve, and how you communicate. It is very difficult to actually go off a tangent and not reach an outcome in the end, whereas in an interviewee-led you always run the risk to investigate some completely irrelevant parts of the issue tree, lose time, then come up with a faulty conclusion.
For these reasons, I believe that you should prepare a set of common skills (structuring, math, exhibit interpretation, communication), which are relevant for both types of cases, however, study and internalize a different game plan and approach for each type of interview.
Then, practice, practice, practice each type of case individually and make it clear at the beginning of each session what the case should be like.
Also, I can recommend you to switch to the interviewer role in some cases. This will give you new and interesting insights into the differences between both approaches.
Let me know if you need any help! I have specialized in the McKinsey interviewer-led format and created a strong and structured approach for each type of question.
So contrary to the advice given here, I would actually advise you to prepare two