Move setting search type from subpages into the search page

[obulat]

Fixes

Fixes #361 by @obulat

Description

This PR moves the detection of search media type, and updating the Vuex Store searchType value from the four separate sub-pages (NuxtChild) to the parent search page. It detects the route.path change, and updates the searchType when necesssary.

Testing Instructions

Run the app, search for 'cat', open the sidebar filters, and watch how it displays the media-specific filters (eg. ImageType for image, Audio Extensions for audio, or nothing for video which is not yet supported) when you change tabs.

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[sarayourfriend]

LGTM! Nice deduplication and clean up.

[zackkrida]

Thank you!