Refactor Vite integration and update docs

Introduced a Vite adapter with "fresh" and "cached" modes, centralized dev server configuration, and streamlined the example setup. Updated documentation with detailed steps for Vite integration and revised the internal dev environment to align with the changes.

docs/mint.json

@@ -89,6 +89,7 @@
         "integration/astro",
         "integration/node",
         "integration/deno",
+        "integration/vite",
         "integration/docker"
       ]
     },