You hear that Meno? (Or is this one of those funny things you were talking about?)
I didn’t make up that funny example myself. I saw it in the Mishnah Berurah. The Mishnah Berurah uses that example regarding the issue of covering your head for mentioning Hashem’s name in general, not
specifically for bentching or davening. It’s clearly two separate issues.
I know these things because I actually looked in the Mishnah Berurah, rather than listing sources that I saw in a footnote or forum somewhere without seeing it inside.