Yes

You are doing it now

Yes, there is. Alpha testing is with a product that’s known to be incomplete, beta testing is with a product which is deemed to be complete, or very nearly so.

Some games with have public alphas, but they typically aren’t AAA titles. It’s usually indie games. 7 Days to Die is a prime example. It was in public alpha testing for several years before they released the “full” game last year. Some indie companies simply can’t afford or don’t have the people for internal alpha testing.