Update lit-starter-ts to lit-next

* Upgrades the source to work with lit-next
* Aligns the linting and formatting rules/settings & ignores with the monorepo
* Upgrades to the same version of typescript and tsconfig settings as the monorepo
* Upgrades the test runner from karma to web-test-runner
* Upgrades the dev server from es-dev-server to @web/dev-server

web-test-runner.config.js

@@ -0,0 +1,39 @@
+import {playwrightLauncher} from '@web/test-runner-playwright';
+import {legacyPlugin} from '@web/dev-server-legacy';
+
+let commandLineBrowsers;
+try {
+  commandLineBrowsers = process.env.BROWSERS?.split(',').map((b) =>
+    playwrightLauncher({product: b})
+  );
+} catch {
+  console.warn(`BROWSER ${process.env.BROWSERS} unknown; using defaults`);
+}
+
+// https://modern-web.dev/docs/test-runner/cli-and-configuration/
+export default {
+  rootDir: '.',
+  files: ['./test/**/*_test.js'],
+  nodeResolve: true,
+  preserveSymlinks: true,
+  browsers: commandLineBrowsers ?? [
+    playwrightLauncher({product: 'chromium'}),
+    playwrightLauncher({product: 'firefox'}),
+    playwrightLauncher({product: 'webkit'}),
+  ],
+  testFramework: {
+    // https://mochajs.org/api/mocha
+    config: {
+      ui: 'tdd',
+    },
+  },
+  plugins: [
+    // Detect browsers without modules (e.g. IE11) and transform to SystemJS
+    // (https://modern-web.dev/docs/dev-server/plugins/legacy/).
+    legacyPlugin({
+      polyfills: {
+        webcomponents: true,
+      },
+    }),
+  ],
+};