context of discussion which includes Christians, their definitions, of G(g)od(s), something pretty central in the Bible, many will automatically refer to their Biblical understanding. If their definition, or a translation of what is actually meant, is not accepted by non believers, then there is the danger people would just, continually, be talking as cross purposes.
Interesting, as that's not my experience. I've had many discussions with Christians where they use the more broad societal definitions in discussing things with non-Christians. It makes sense to have these discussions using broadly understood and value-neutral terminology where possible.
Because we weren't just discussing Christians or Christianity, were we? And it's a wee bit arrogant to assume that a discussion about religion or lack thereof should take place using the terminology of Christian doctrine when there are perfectly good and well defined terms out there that won't lead to the confusion we've experienced.
Why must non-Christians accept your definitions? Why aren't you prepared to bend the other way?