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

docs-src/api.11ty.cjs

@@ -15,7 +15,9 @@ module.exports = class Docs {
     const tags = customElements.tags;
     return `
       <h1>API</h1>
-      ${tags.map((tag) => `
+      ${tags
+        .map(
+          (tag) => `
         <h2>&lt;${tag.name}></h2>
         <div>
           ${tag.description}
@@ -23,41 +25,40 @@ module.exports = class Docs {
         ${renderTable(
           'Attributes',
           ['name', 'description', 'type', 'default'],
-          tag.attributes)}
+          tag.attributes
+        )}
         ${renderTable(
           'Properties',
           ['name', 'attribute', 'description', 'type', 'default'],
-          tag.properties)}  
-        ${/*
+          tag.properties
+        )}  
+        ${
+          /*
            * Methods are not output by web-component-analyzer yet (a bug), so
            * this is a placeholder so that at least _something_ will be output
            * when that is fixed, and element maintainers will hopefully have a
            * signal to update this file to add the neccessary columns.
            */
-          renderTable(
-          'Methods',
-          ['name', 'description'],
-          tag.methods)}
-        ${renderTable(
-          'Events',
-          ['name', 'description'],
-          tag.events)}    
-        ${renderTable(
-          'Slots',
-          ['name', 'description'],
-          tag.slots)}  
+          renderTable('Methods', ['name', 'description'], tag.methods)
+        }
+        ${renderTable('Events', ['name', 'description'], tag.events)}    
+        ${renderTable('Slots', ['name', 'description'], tag.slots)}  
         ${renderTable(
           'CSS Shadow Parts',
           ['name', 'description'],
-          tag.cssParts)}
+          tag.cssParts
+        )}
         ${renderTable(
           'CSS Custom Properties',
           ['name', 'description'],
-          tag.cssProperties)}
-        `).join('')}
+          tag.cssProperties
+        )}
+        `
+        )
+        .join('')}
     `;
   }
-}
+};
 
 /**
  * Renders a table of data, plucking the given properties from each item in
@@ -65,7 +66,7 @@ module.exports = class Docs {
  */
 const renderTable = (name, properties, data) => {
   if (data === undefined) {
-    return ''
+    return '';
   }
   return `
     <h3>${name}</h3>
@@ -73,13 +74,17 @@ const renderTable = (name, properties, data) => {
       <tr>
         ${properties.map((p) => `<th>${capitalize(p)}</th>`).join('')}
       </tr>
-      ${data.map((i) => `
+      ${data
+        .map(
+          (i) => `
         <tr>
           ${properties.map((p) => `<td>${i[p]}</td>`).join('')}
         </tr>
-      `).join('')}
+      `
+        )
+        .join('')}
     </table>
-  `
+  `;
 };
 
 const capitalize = (s) => s[0].toUpperCase() + s.substring(1);