I think you've got the right idea, though in practice the initial "authentication" question (you are who you say you are) is very closely linked to the initial "authorization" evaluation (can you enter).... because in most systems the only "can you enter" authorization required for access is in fact that you are who you say you are.

But not all systems work this way. There are some systems where you can log in successfully, but then are immediately escorted out because the "can you enter" question has secondary considerations or is decided once identity has been established based on a larger criteria. Expired accounts in some systems work exactly like this.

One problem is that treating authentication as a "can you enter" authorization is predicated on the idea of a session-based system with two states, logged in or logged out. But there are many scenarios where e.g. taking some particular action requires authn and authz, regardless of login status. A simple example is performing some destructive action.

The distinction between authentication and authorization allows modeling of many different kind of systems, including the degenerate case where identification is treated as a proxy for authorization.

Btw, the kind of thinking behind that degenerate case is what leads to IDOR security bugs - "this person is logged in, so they can access whatever the URL says... even if it's another customer's data!" It turns out that thinking clearly about security helps be more secure, and unfortunately, vice versa.