refactor(tui): Move examples from README to test directory

- Move chat application example to test/chat-app.ts
- Move multi-component layout example to test/multi-layout.ts
- Update README to reference example files instead of inline code
- Add proper exit handlers (Ctrl+C) to all examples
- Simplify README by reducing inline code examples

packages/tui/test/multi-layout.ts

@@ -0,0 +1,85 @@
+#!/usr/bin/env npx tsx
+import { TUI, Container, TextComponent, TextEditor, MarkdownComponent } from "../src/index.js";
+
+/**
+ * Multi-Component Layout Demo
+ * 
+ * Demonstrates:
+ * - Complex layout with multiple containers
+ * - Header, sidebar, main content, and footer areas
+ * - Mixing static and dynamic components
+ * - Debug logging configuration
+ */
+
+const ui = new TUI();
+
+// Create layout containers
+const header = new TextComponent("📝 Advanced TUI Demo", { bottom: 1 });
+const mainContent = new Container();
+const sidebar = new Container();
+const footer = new TextComponent("Press Ctrl+C to exit", { top: 1 });
+
+// Sidebar content
+sidebar.addChild(new TextComponent("📁 Files:", { bottom: 1 }));
+sidebar.addChild(new TextComponent("- config.json"));
+sidebar.addChild(new TextComponent("- README.md"));
+sidebar.addChild(new TextComponent("- package.json"));
+
+// Main content area
+const chatArea = new Container();
+const inputArea = new TextEditor();
+
+// Add welcome message
+chatArea.addChild(
+	new MarkdownComponent(`
+# Welcome to the TUI Demo
+
+This demonstrates multiple components working together:
+
+- **Header**: Static title with padding
+- **Sidebar**: File list (simulated)
+- **Chat Area**: Scrollable message history
+- **Input**: Interactive text editor
+- **Footer**: Status information
+
+Try typing a message and pressing Enter!
+`),
+);
+
+inputArea.onSubmit = (text) => {
+	if (text.trim()) {
+		const message = new MarkdownComponent(`
+**${new Date().toLocaleTimeString()}:** ${text}
+		`);
+		chatArea.addChild(message);
+		ui.requestRender();
+	}
+};
+
+// Build layout
+mainContent.addChild(chatArea);
+mainContent.addChild(inputArea);
+
+ui.addChild(header);
+ui.addChild(mainContent);
+ui.addChild(footer);
+ui.setFocus(inputArea);
+
+// Configure debug logging
+ui.configureLogging({
+	enabled: true,
+	level: "info",
+	logFile: "tui-debug.log",
+});
+
+// Handle Ctrl+C to exit
+ui.onGlobalKeyPress = (data: string) => {
+	if (data === "\x03") {
+		ui.stop();
+		console.log("\nMulti-layout demo exited");
+		process.exit(0);
+	}
+	return true;
+};
+
+ui.start();