chore: delete `api-routes` example completely (#72437)
## Description
At #68771,
[`api-routes`](https://github.com/vercel/next.js/tree/canary/examples/api-routes)
example was renamed
[`route-handlers`](https://github.com/vercel/next.js/tree/canary/examples/route-handlers).
However,
[`.gitignore`](https://github.com/vercel/next.js/blob/canary/examples/api-routes/.gitignore)
wasn't deleted and a new
[`.gitignore`](https://github.com/vercel/next.js/blob/canary/examples/route-handlers/.gitignore)
file isn't updated.
In addition, some `api-routes` names exist in the other place.
That's why I update those.
### Improving Documentation
- [x] Run `pnpm prettier-fix` to fix formatting issues before opening
the PR.
- [x] Read the Docs Contribution Guide to ensure your contribution
follows the docs guidelines:
https://nextjs.org/docs/community/contribution-guide
Co-authored-by: JJ Kasper <jj@jjsweb.site>
