“Can you design it exactly like [the competitor’s website]? No, I mean, exactly like it?”
To which the proper response should be: "Do you know the penalty for copyright infringement? Because that is what you are discussing and I want to state now, for the record, in case any lawyers are reading this in the future that I had no part in any copying of a competitor's website or their content."
It is okay for clients to want to emulate a website that they admire. But they can never duplicate a site completely, word for word, code for code. That is grounds for a lawsuit. Copyright infringement in Canada can lead to fines of up to $1,000,000 CDN plus any civil lawsuits from the company which was infringed.
Beyond that the thing to do is ask the client what specific elements of the website they really like. Get examples of things they like and learn exactly what aspects of it they like best.
Also ask them what they would change to make it better.
Once those things are nailed down the next step should be to determine a layout. For this I prefer to used old school pen and paper so that we have a hard copy of what the layout design is.
I also sometimes ask a client to "sign off" on a finished layout by initializing the corner of the draft design that was completed on paper. This way they know they have just agreed to this layout/design and this is the layout/design that will be made. Changes may be made later, but any amendments should be relatively minor.
Any major changes imply a whole new design, possibly starting completely fresh. Which means additional costs in terms of time and money.