The second quote is about stochastically determining which side is the "head" and which is the "tail", effectively, which occurs by positive feedback of an existing gradient. This is not really a matter of information stored in the sense that we think of information stored in DNA bases or bits on magnetic tape, the passage seems to be referring to information in the sense of a reduction in uncertainty about the state of the universe. The way a developing embryo chooses where the head is going to be is like dumping a bag of M&Ms on the table and having the head develop wherever the highest concentration of yellow M&Ms are. If you expect them to be uniformly distributed, you'd be correct statistically, but for any one iteration there will be a non-uniformity that determines what happens next: that's information, reduction in statistical uncertainty. If you, via the power of a thought experiment cousin of Maxwell's demon, were to scramble the positions of molecules in the egg cell, that would change where the "head" develops.
Yet, all of this is guided by proteins that are created and function according to the "instructions" in DNA. DNA doesn't have information about where in the future there will randomly be a concentration of yellow M&Ms, but it does have the information to synthesize proteins that react to concentrations of yellow M&Ms, causing a positive feedback loop that further concentrates them on a particular axis.
For embryonic development, these yellow M&Ms are mRNA transcripts, and at least in drosophila, these initial transcripts are indeed made in the egg cell, not post-fertilization (see https://en.wikipedia.org/wiki/Drosophila_embryogenesis#Anterior-posterior_axis_patterning_in_Drosophila). It's possible to introduce a mutation to a fly that will only affect her offspring, by interfering with transcription of these mRNAs. More generally, maternal effects are observed when the egg cell or developing environment has some influence on the offspring.
However, all those influences are coded in the DNA of the next generation, too, so it also seems reasonable to say that the genome contains all the information necessary. Containing all needed information to do X is not the same as being able to do X. To turn DNA into RNA transcripts, and to turn transcripts into proteins, requires more than information, it requires machinery. We can say the information to build those machines is also in the DNA.
Honorable mention to mitochondrial DNA, which comes entirely from the egg cell.
Generally, I think learning about biology from these grand pronouncements like "all information is in DNA" is a bad way to learn biology. Biology is not mathematics, we do not operate in the certainty of proof. Learn how things actually work, first, rather than trying to prove or disprove "truths". These statements are meant to capture wonder and convey the basic order of things, they are not dogma (even if someone regretfully calls it that).