In the one non-open-source company I worked at, the first-party API was an extension of the public API. Some methods took additional internal-only parameters if you were using one of the official app IDs, for example. Some returned additional fields in responses. A bunch of additional internal-only methods also became available.

I mean, if you have a public API, it makes sense, engineering-wise, to build your internal API on top of that, doesn't it? But apparently not everyone does it like this.