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You are correct that the first statement needs no comma before rather. Here, the expression rather than [to] a restaurant is essential information for understanding the statement. It also describes or explains grocery store, again indicating it's importance.
Commas separate parts of sentences. Because you don't want to separate the final phrase in the first example, you don't use a comma.
In the second example, rather than going out to a restaurant, you still don't need a comma before rather. Here, the expression also provides necessary information, as in the first case. The phrase is not parenthetical, and it certainly isn't an appositive.
However, you will need to follow the expression with a comma because it is serving as an introductory dependent phrase, as in "Rather than going to the store, we went to the restaurant."
But why no comma before rather in the second example? The word that turns the following expression into a noun phrase, here to be used as the direct object of decided. If we place a comma after that, we separate the expression from the noun phrase, which is not correct because it needs to be part of the noun phrase.
Bottom Line:
First example: We decided to go to the grocery store rather than to a restaurant.
Second example: We decided that rather than going out to a restaurant, we would go to the grocery store.
You might pick up a copy of Zen Comma, which has a much more thorough discussion of comma uses.
On a side note: You seem to be confused about appositive phrases. Although appositives don't provide essential information, not every non-essential phrase is an appositive. I think you mean parenthetical expressions, of which appositives are one type, or non-restrictive phrases and clauses.
Example appositive: "This toy, a 1992 Barbie doll, is a family treasure." A 1992 Barbie doll is an appositive.
Example non-restrictive clause: "Take away my life, which is as precious to me, but don't take my dignity." Which is precious to me is the non-restrictive clause.
The primary difference in the forms of the two sentences is that the first one has "eats... reading..." and the second one has "was making... talking..." It thus seems that the first sentence—punctuation aside—can be read only as signifying
He always eats breakfast [while] reading the paper.
whereas the second sentence—again, ignoring punctuation—can be read as meaning either
She was making sandwiches [while] talking with her daughter.
or
She was making sandwiches [and] talking with her daughter.
So perhaps the comma is there to signal a dropped and and to emphasize the parallelism between "making sandwiches" and "talking with her daughter," both of which attach to the word was if an implied and is at work. I certainly don't see the argument for putting a comma after "sandwiches" in the second sentence as being any stronger than the argument for putting one after "breakfast" in the first, if the point is simply to mark the absence of the intended word while in both cases.
Still, I'm at a disadvantage in trying to work out the book's reasoning because my preference would be to include the missing while in the first sentence and the missing while or and in the second sentence—and if you do those things, neither sentence needs a comma.
Best Answer
By including the comma in the example sentence, the author clearly packages "in part" with "because" instead of with "treadmill." Admittedly, the ambiguity isn't as great in your example as it may be in some others, but including the comma still seems to me to be at least somewhat helpful.
For a more problematic example, consider this one:
Reading this sentence in a world where commas were not permitted before "because" or before "in part because," I would have no way of knowing (from the sentence) whether the whole estate is supposed to go to the housekeeper (for reasons that include, among others, the one stated) or whether part of the estate is supposed to go to the housekeeper (for the sole reason given). In a world where commas are permitted before "because" and before "in part because," the author can make his or her intention much clearer by adding a comma in the appropriate location:
if the housekeeper is supposed to get everything, and
if the housekeeper is supposed to get something.
(The statement might still be deemed ambiguous, since someone advocating on behalf of the housekeeper might argue that "housekeeper in part" is a self-contained phrase—in effect, "housekeeper-in-part"—and that the intention of the testator was to give all of the estate to the "housekeeper in part." This sort of complication explains why lawyers will always be with us.)