There are several different definitions. One definition I like is that alpha testing is when you know there are still missing features, while beta testing is when the planned features exist and you're just trying to make sure that they all work properly -- in which case you could very easily have public alpha testing.
To find those more 1k bugs to fix I would need to open it to some users. We are testing it almost everyday and now are unable to uncover more bugs. I mean bugs will always be there. Windows XP has lot of bugs. As I am advised by others, at one point if you feel codes is stable enough, you need to open it to get more feedback.
* blink *
How much code do you have? I've been writing code for my own startup for about 20 months, and after 25k lines of code I'd say I've fixed maybe 50 bugs in total.