A gene is a unit of heredity.
The genome is the sum total of all genetic information in an organism.
Both are extremely loose terms. Individual sub-fields of biology/biochemistry may use more specific variations in specific contexts, but the definitions above cover all of those meanings.
(I.e. per my sibling post, there were jargon terms with these names that had one particular definition under an early, black-box mental model of heritability that is no longer used/favored; and there are new replacement jargon terms with the same names, that have a different definition under the more-modern mental model of germline heritability.)
After all, there are entirely non-genetic mechanisms of heredity. You inherit some of your mother's immune histochemistry through developmental exposure, and more through consuming breastmilk. Yet there is nothing that "codes" for this histochemistry; it's just a stateful process (mother's immune system) interacting with another stateful process (foetal immune system) — working almost more like a "colonization" of commensal bacteria than like gene transfer.
In terms of "heritable traits" — "nature vs nurture" — such effects appear squarely on the "nature" side. But they're not germline heritable. Implant the fertilized egg into a surrogate mother, and you get different histochemistry. Have a wetnurse feed the baby, and you get different histochemistry.
If these things are "genetics" — if the information carried by the state of immune cells in response to observing natal antigens is "genes" — then the term "genetics" has no useful meaning / is useless to talk about the thing we want to talk about when we talk about genetics. Where "what we talk about when we talk about genetics", is how an organism's breeding/germline ancestry, predicts phenotypic outcomes.
And if you agree that that is the thing we want to talk about — then epigenetic information isn't "genes" (under the germline-heritability model) either. Your epigenetics is hereditary, but only in the same sense that your immunohistochemistry is hereditary — being passed via exposure in a natal environment, rather than being "written into" the fertilized egg.
(Also — at least AFAIK — we don't even consider our nuclear mitochondrial DNA to be a part of our genomes. Even though it is passed through the germline, and is intensely important in many congenital metabolic dysfunctions. Because our mitochondria are not technically "us" — we might call them organelles, but they're still more like commensal prokaryotes in how cells treat them: dividing on a separate timeline from our cells; not assortatively assigned to daughter cells by spindle fibers during mitosis; not observed + checkpointed to ensure sufficient numbers exist for daughter cells before telophase; etc.)