I interpret the "To Be or Not to Be" soliloquy as actually presenting three options:
1. "To Be" - That is to quietly "suffer the slings and arrows of outrageous fortune"
2. "Not to Be" - That is to "take arms against a sea of troubles"
3. "To die, to sleep" - that's when I think he first actually starts talking about suicide.
I think when Hamlet says "to be or not to be" he's not actually using "be" in a way that means "to exist or not to exist" (at least not at that point, at the beginning of the soliloquy), but instead the same way that Paul McCartney uses it in "Let it Be".
Also, one might be able to reverse the meanings I've attached to "To Be" and "Not to Be" and argue that he is saying, by refusing to act, he is denying his own existence.
The real greatness in many pieces of Shakespeare is that they are written in a way where they can be interpreted in many different ways, for many different reasons. The way I feel a line could be the same as originally intended, or it could be something completely different. I would guess it is based on many factors including personal experience, culture, education, and creativity. That is why I am a fan of language.