What's sort of funny here in a web forum is that people usually keep their true identities
private (or maybe your name really is Catlady?
) ...but are open about Buddhism.
Personally, I have always felt a little self-conscious in non-buddhist situations about openly being Buddhist. So, I guess even if there is no self, one can still feel self-conscious. But this really has more to do with the fear that others will really not understand what (my) Buddhism is, and will misinterpret it, and will assume a lot of things about me or what i do which are not really true.
So then, this brings up the issue of why that even matters.
Of course, as you mention, in business it might matter. If you run a business in an area where people are rather closed-minded or where they themselves feel threatened by what they don't understand, and they thought all sorts of negative things about Buddhists, then keeping things to yourself would not be a bad idea.
I make a distinction between being untruthful and guarding the truth
. Because, the word "Buddhism" is just a label and people can stick that label on anything they imagine it fits on.
So, for example, if a person thinks Buddhists are cannibals, and you tell that person you are a buddhist, then, from their point of view you are telling them that you are a cannibal. To me, this is almost like lying to the person because you may be reinforcing some misinformation that they have. On the other hand, when they see that you are not going to cook and eat them, this opens up the opportunity to get rid of bad assumptions.
A couple of years ago, some carpenters came to my house to do some renovation work. One of them noticed my (Tibetan style) Buddhist shrine, with offering bowls filled with water in front. He said, "Are you a buddhist? I know a Vietnamese guy who is a Buddhist . He feeds his Buddha a plate of food every day. "What do you feed your Buddha?"
I was completely knocked for a loop. Speechless.
"uhhhhh....ummmmm....mine just drinks water." I told him.
To this day, I regard "What do you feed your Buddha?"
as a great teaching.