I was listening to a Daniel Glass podcast and I was able to relate to a lot of what he had to say. The gist of his message was that a drummer should at times be part actor. By that he means that when playing, for example, a cover tune, the drummer may have some responsibility to play it 'just like the record'. I can't imagine playing a Jackson Browne tune and not trying to channel Russ Kunkle at least a little bit! Daniel goes on to make an argument for reframing the situation as an opportunity.
I have my own interpretation of this. You see, I don't really want to play like the drummer on the record. That's not why I play. I play to add my spin to the music, and I can't do that if I'm copying someone else. Copying is a very important part of the learning process, but to copy someone else's style as a matter of course to me seems contrary to the whole business of making music. I rarely play exactly what was on the recording. I prefer to emulate the 'spirit' of the original artist and use that as inspiration for my own interpretation.
In a cover band, you're not necessarily obligated to ape the original drummer. Still, the audience is expecting to hear their favourite tunes the way they've come to know them, so things ought to sound more-or-less correct in the drum department. When you're playing a well-recognized tune -- especially one with 'signature' drum parts -- then I think we have an obligation to pay homage to the original.
In a tribute-style band, the goal is to replicate the experience of the original band. That means the tunes, the outfits, the stage presentation ... everything … should be true to that experience. And so the drum parts need to be virtually identical to the original. But even then, there can be room for your personal touch.
What Daniel suggests, and I fully endorse, is that you step inside the other artist's skin for a few minutes. He even goes as far as to suggest you imagine yourself as that drummer to help you emulate their style and energy. The goal is not to replicate the drummer, but to approach the tune the way you think they might. And when a chance for a cool groove or fill comes up, you can ask yourself, "What would Daniel Glass do?"