All Plex types (browse cards, show/season/item details, collections, native
playlists, filters/facets, watch-link) + EMPTY_PLEX_FILTERS/plexFilterCount/
appendPlexFilters + the ~40 plex methods move into a self-contained api/plex.ts
exporting a `plexApi` slice (imports only core's req/beacon). api.ts spreads the
slice and re-exports the types via `export *`. api.ts 1352 -> 882 lines (halved
from the original 1793); no runtime change. Gate green: typecheck, eslint,
prettier, 74 vitest.