Yes, just use. The fact that less common edge cases exist doesn't undermine the solution. The example in the article was example.com and api.example.com, the most common setup imo.

> What if the API is shared between multiple domains? Do you need to reverse proxy it everywhere?

It would depend on your specific setup. We use a single nginx server entry to proxy many domains.

> What if it is a public API for third-party sites?

Then this approach would not be viable.