bina – this is what pba said that I was addressing:
“We should tell kids to behave better than normal because the torah teaches us to.
I feel like we’re telling kids to behave better than normal because we want people to think the torah teaches us to.”
He seems not to have a problem with telling kids to behave better than normal, but rather with the reasoning behind it — i.e. to project an image rather than because it is required by the Torah.
As far as what is “normal” for kids to do, if it is age appropriate behavior there probably is not an issue of Chillul Hashem, as there is nothing wrong being done. However if in fact your kids are capable of behaving better then normal, I can definitely hear that is good Chinuch in this general concept of people respecting Torah via the positive behaviors of those who keep it. But I would try to explain it more as I said above — something to the effect that the Torah requires us to always behave with Derech Eretz, speak respectfully, not fight, etc. When we are in public and people know that we are frum Jews we have another special responsibility to show everyone what the Torah teaches us, and be extra careful to follow everything we are taught.
I don’t know why I alway seem to resort to kids songs on these hashkific threads, but the chorus of the Kiddush Hashem song on the Marvelous Middos machine goes something like “you’ve got to show the world what the Torah teaches you” — that’s the message I try to send to my kids, not that we are putting on a show to give a false impression.
Again all this is dealing the with the third concept, not the actual main definitions of Chillul Hashem.
Taking welfare if it would fit in altogether would be in the second category (something that causes others to look down on you), which is actually more within the Halachic parameters of Chillul Hashem. But it is not clear that it would apply, as a big part of the question is if it is morally/ethically wrong to take welfare money or not (assuming you legitimately qualify, obviously). If it’s not wrong, but only gets people annoyed or doesn’t fit in with an individual’s work ethic, it is not likely a Chillul Hashem. As with all areas of halach lmaasah, everyone should ask their own Shaalah!