Split search into shared state with inline homepage input and modal

src/lib/searchModal.svelte.ts

@@ -1,13 +1,26 @@
 let open = $state(false);
 
+// When a page hosts an inline search input (the homepage), it registers a
+// focus function here and every search trigger routes to it instead of the
+// modal, so a modal input never opens on top of an inline one.
+let inlineFocus: (() => void) | null = null;
+
 export const searchModal = {
   get open() {
     return open;
   },
 };
 
+export function registerInlineSearch(focus: () => void): () => void {
+  inlineFocus = focus;
+  return () => {
+    if (inlineFocus === focus) inlineFocus = null;
+  };
+}
+
 export function openSearch() {
-  open = true;
+  if (inlineFocus) inlineFocus();
+  else open = true;
 }
 
 export function closeSearch() {