SearchController refactor some instance variables

What does this MR do and why?

Related to #378012 (closed)

Remove two instance variables from the SearchController only used in HAML and move them into the SearchServicePresenter.

Update specs (including adding feature_category)

Screenshots or screen recordings

N/A

How to set up and validate locally

Numbered steps to set up and validate the change are strongly suggested.

MR acceptance checklist

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