I believe I found – B”H – the place where I remember Lubavitch scholar Simon Jacobson discussing the issue that started this thread, and, as stated previously, he says that it is * not * to be understood as literally “a piece of G-d”.
In Episode 13 of his “Tanya Applied with Rabbi Simon Jacobson” (available at YouTube), posted a little over three months ago, at just after 19:45 he states “obviously we are not talking about a piece of G-d”.
(This is not an endorsement of learning Tanya, or his talks in general, just showing that even a leading Chabad Lubavitch scholar is in accord with what tiawd posted at the beginning of this thread on this point.