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?
I could be in some sort of bubble, but hardly can remember the case when dynamic part of request has not flown through some sort of reverse proxy or fully capable webserver (like Apache) anyways. On dev envs of course seen, but not in production.
> 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.