Is Publons a reliable reference to show peer-review experience in a faculty application?
Then, I felt it was better if I did not mention Publons at all (it
  would look more normal). Was it a problem of my interviewer OR it is
  not common to include a peer-review profile in the application?


I have seen a number of people (myself included) include a peer-review profile. In my case, it simply lists the journals I have reviewed for. At this point, it's actually more the list of journals I frequently review for.

I think the issue is more with Publons - in my experience, there hasn't really been a strong need for tracking credits the way they want, and I found it was more hassle than it was worth to keep it up-to-date (as if you have a Publons profile and highlight it, the implication is that it's accurate).

Essentially, I trust what you've told me. Because specific reviews, rather than the abstract concept of being an active reviewer, isn't really much of a consideration for me. It's a nice thing, and I appreciate seeing it mentioned as I do consider it a professional obligation, but Publons inherently wants there to be a quantitative impact based on volume and I...simply can't be bothered.