docs: extract inline code examples into standalone projects

Extract inline @code blocks from vt headers (size_report.h, modes.h,
sgr.h, paste.h, mouse.h, key.h) into standalone buildable examples
under example/. Each header now uses Doxygen @snippet tags to include
code from the example source files, keeping documentation in sync
with code that is verified to compile and run.

New example projects: c-vt-size-report and c-vt-modes. Existing
examples (c-vt-sgr, c-vt-paste, c-vt-mouse-encode, c-vt-key-encode)
gain snippet markers so their code can be referenced from the headers.
Conceptual snippets in key.h, mouse.h, and key/encoder.h that show
terminal-state usage patterns remain inline since they cannot be
compiled standalone.

example/c-vt-modes/src/main.c

@@ -0,0 +1,45 @@
+#include <stdio.h>
+#include <ghostty/vt.h>
+
+//! [modes-pack-unpack]
+void modes_example() {
+  // Create a mode for DEC mode 25 (cursor visible)
+  GhosttyMode tag = ghostty_mode_new(25, false);
+  printf("value=%u ansi=%d packed=0x%04x\n",
+      ghostty_mode_value(tag),
+      ghostty_mode_ansi(tag),
+      tag);
+
+  // Create a mode for ANSI mode 4 (insert mode)
+  GhosttyMode ansi_tag = ghostty_mode_new(4, true);
+  printf("value=%u ansi=%d packed=0x%04x\n",
+      ghostty_mode_value(ansi_tag),
+      ghostty_mode_ansi(ansi_tag),
+      ansi_tag);
+}
+//! [modes-pack-unpack]
+
+//! [modes-decrpm]
+void decrpm_example() {
+  char buf[32];
+  size_t written = 0;
+
+  // Encode a report that DEC mode 25 (cursor visible) is set
+  GhosttyResult result = ghostty_mode_report_encode(
+      GHOSTTY_MODE_CURSOR_VISIBLE,
+      GHOSTTY_MODE_REPORT_SET,
+      buf, sizeof(buf), &written);
+
+  if (result == GHOSTTY_SUCCESS) {
+    printf("Encoded %zu bytes: ", written);
+    fwrite(buf, 1, written, stdout);
+    printf("\n");  // prints: ESC[?25;1$y
+  }
+}
+//! [modes-decrpm]
+
+int main() {
+  modes_example();
+  decrpm_example();
+  return 0;
+}