This way of thinking works for Buddhists, but I wouldn't advise telling people who don't understand karma to practice like this.
This sounds weird. Why should a wholesome way of thinking work for Buddhists only? And how could someone who does not understand karma think correctly about it? Besides, how could someone who thinks correctly about karma fail to understand it? Just because someone does not label oneself a “Buddhist”?
Not only the quoted statement is illogical, but also it lacks any Buddhist lineage. Maybe Thubten Chodron, born as a Jewess, just expressed deeply ingrained non-Buddhist, Jewish atavistic preconceptions, according to which some Jewish precepts apply exclusively to the Jew.
For instance, while Jewish teachings forbid killing, this applies only to another Jew. The Jewish Torah (roughly equivalent to the Christian Old Testament) is filled with examples of mass-murderings of innocent non-Jewish women and children being presented as “virtuous”.
Also, Jewish law does not allow a Jew to charge interests on a loan to another Jew, but allows a Jew to charge interests on a loan to a non-Jew. This is because Jews see themselves as intrinsically different from non-Jews, as a “chosen people” subject to special rules and precepts.
However, such preconceptions do not apply to Buddhist teachings. According to Buddhism, every single sentient being, even the most insignificant insect, partakes of the Buddha's lineage. Everyone is capable of understanding karma, and of improving their lives through such understanding, irrespectively of labeling or not oneself as a “Buddhist”.
Buddhism is all-embracing, open, non-exclusivistic. And one should be wary of mixing pure Buddhist teachings with illogical garbage, lacking any Buddhist lineage.