A faculty interview is short, whether it be a skype one or an onsite one. Esp for the former, as interviewers have tight and non-negotiable time limits, it's so short that there's little space beyond answering the interviewers' direct questions. In a short interview, as one invariably improvises, no matter how well they were prepared, it seems. Many rehearsed lines (not necessarily the exact wording, but topics planned to highlight) are forgotten. How to bring into full play what was prepared?
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Don't overthink it.
In a skype interview, you're right, it is mostly their questions. Answer them directly, and maybe keep 2 points in mind that you really want to make when they give you a chance to lead the conversation for a few minutes.
For on-site interviews, it's a long day or two, and you'll have plenty of opportunities to say the things you want to say.
I suggest you trust that the important things will come to light during the interview. If you forget one or two things you wanted to say, that omission is very unlikely to determine the success or failure of your interview. The overall picture that the department gets of you is much more important -- is your research strong? does it fit well with the department? will you be a good teacher? a good colleague? I suspect that the "rehearsed lines" will be less important, in the scheme of things.
"In a short interview, as one invariably improvises, no matter how well they were prepared, it seems."
I disagree. If you practice enough, you can give prepared answers. Just like any other kind of performance, interviewing needs to be practiced. More practice will lead to better results.