Version: 2.19.0 (also present on main)

Describe the bug

In any media-type field, clicking Select Media opens the picker fine, but
typing in its search box (a) doesn't filter the results, and (b) renders a whole
new search-box-plus-grid panel inside the grid — nesting one level deeper
with every keystroke.

Root cause

Two defects in GET /admin/media/selector (packages/core/src/routes/admin-media.ts):

1. The search input has an id but no name, yet relies on
   hx-include="[name='search']" — so the typed term is never sent:

   <input type="search" id="media-selector-search" …
          hx-target="#media-selector-grid"
          hx-include="[name='search']">   <!-- nothing on the page is named "search" -->

2. The endpoint always returns the full panel (search box +
   #media-selector-grid). Since the input's hx-target is the inner grid,
   htmx swaps that entire panel into the grid → a nested panel per keystroke.

To reproduce

1. Edit any content type that has a media field.
2. Click Select Media.
3. Type in the picker's search box.
4. Results don't filter; a nested search-box+grid appears inside the grid and
   deepens with each keystroke.

Expected

Typing filters the grid in place; the search box stays put (single, not nested);
selecting a file still fills the field.

Proposed fix

1. Add name="search" to the input so hx-include sends the term.
2. Return a grid-only fragment for the HTMX search request (detected via the
   HX-Target: media-selector-grid header) and the full panel only on the
   initial modal load.

I have a working fix with a regression test and can open a PR.