Converted `api-routes-rate-limit` example to TypeScript to match Contribution docs.
- Removed dead/non-standard/typo links for "Deploy You Own ▲" and "View Source"
## Documentation / Examples
- [X] Make sure the linting passes by running `pnpm lint`
- [X] The examples guidelines are followed from [our contributing doc](https://github.com/vercel/next.js/blob/canary/contributing.md#adding-examples)