Knowledge and understanding of the "classic" strategy/marketing/ops frameworks is important since usually they cover or the most important and common drivers and therefore allow not to "reinvent the wheel" everytime you have a problem in front of you (they can also help if there is interview fright that is blocking or slowing thought process)
However it is key to customize them according to the situation by either adding or dropping some of the drivers or emphasizing some more than others depending on the case prompt that was given (e.g. if in prompt interviewer says to ignore pricing because it's fixed, then do not cite price in your driver analysis).
In terms of renaming I would say I wouldn't not go through the effort of doing so unless names used during the case for the driver are different (e.g. instead of place, the interviewer talks about channels, then use the word channel).
One thing I would strongly suggest to not announce "I am going to use so and so framework" but be more natural about it "I would like to explore the following drivers". Both you and interviewer will recognize that a framework is used but it doesn't come across as you are trying to force-fit a solution to a custom problem.
Hope this helps,