The separation of church and state

Over at Ragged Glory, I have posted the following question:  What do Americans mean when they refer to the separation of church and state? I can think of two ways that the phrase might be interpreted:

1. US Presidents should make no reference whatsoever to God while speaking in their official capacity.
2. It is OK for US Presidents to refer to God, but only in generic terms. They should not express a preference for any specific religion or understanding of God.

I am not an American, and I haven't studied the subject, so I am inviting you (even if you are not an American) to educate me.

What do you understand the phrase to mean?