Nice question! Let's start by talking about different forms of heritability.
What is heritable?
Here is a list of things that can be inherited. Some of them are only ('mainly' would be more correct that 'only') transmitted to the offspring while others can also be transmitted to any individual in the population.
The term epigenetics is used with several meaning. In its narrowest sense it refers only to modifications (typically methylation) on top of the DNA sequence or on top of histones (histones are proteins around which the DNA is wrapped). In its broadest sense, it refers to any element of hererdity that is not caused by the DNA sequence. In the below list, I will avoid using the term epigenetics.
What can be inherited:
- DNA (obviously)
- Modification of the sequence of DNA
- Methylation on DNA
- Modification of (typically) methyl group added on nucleobasis
- Methylation on histone
- Modification of (typically) methyl group added on (typically) the tails of the histones.
- Environment (or rather macro-environment)
- The term environment can be used with different definitions. Histone modification is generally considered as being the environment of the genome typically. I will however make a (somewhat) arbitrary distinction between this micro-environment and the more intuitive macro-environment
- Individuals affect their environment and therefore can shape the environment for their descendents. An obvious example is a beaver dam which last longer the the life of a single beaver. Two key concepts here are niche construction and Ecosystem engineering. You should have a look at the post Does modern theory of evolution include modification of physical environment?
- Memetics (See wikipedia > meme)
- A meme is any idea or element of culture that can be transmitted. This type of inheritence is particularly important is species that have high cognitive abilites such as humans.
In this kind of discussion, there is often someone who stand up to talk about plasticity, developmental noise and mutational noise to mess things up. I will just consider them as phenotypic trait and will not much attention to them for the moment (even though part of my research is focused on these three concepts).
Consideration of the non-genetic inheritence in the literature
I like that you noticed that "the tag genomes is defined as The
entirety of an organism's hereditary information". Such definitions are common and grounded in an old view of inheritance. As listed above, there are other things than the DNA sequence that is inherited by the offspring. While all biologists know that, it is true that few really acknowledge the existence of these other forms of inheritence in their daily thoughts.
Some authors have argued that these other types of inheritence must not be neglected. Odling-Smee and colleagues wrote a whole book on the subject (Niche Construction: The Neglected Process in Evolution). Some have even argued that we should rename our theory of evolution to the light of the existence of non-genetic inheritence (see Laland et al. 2015). My feeling is the vast majority of biologist recognize the existence of these types of inheritence but do not consider that it is worth a renaming of the theory of evolution (current named modern evolutionary synthesis). Some would not consider this renaming necessary either because they neglect the important of these other forms of inheritence or because they just don't think it is worth renaming our theory even in the recognition of the importance of these other processes.
Viewing the cell as a decoder can be misleading
In your question, you are suggesting to view the cell as a decoder of DNA. IMO, this is misleading. In this discussion, I will exit science and enter in the realm of philosophy (identity).
What is generally considered as the mechanism of decoding DNA is actually part of our geome itself (consider typically the importance of tRNA and rRNA). If uneasy with basic concept of gene expression you might want to have a look at an intro course such as Khan Academy from the section
Classical and molecular genetics to the section
Also, I would argue that all the elements that are being transmitted to the offspring shall be considered as being the offspring. If it turns out (which we actually pretty much know is wrong) that histone modification explains more in the phenotypic variance than genetic variance, it would intuitively be unfair to see histone modification as just being a decoder of the genome. We would rather be tempting to see histone modification as central and the DNA sequence as a side-effect.
You say "This question seems never to be seriously addressed in lay literature". Richard Dawkins book called the extended phenotype, while maybe not directly addressing your question does a good job at playing out with thee concept of environment vs gene separation. You might want to have a look at it.
Note that the laland et al. 2015 cited above is easy to read and would be a good source of information for a layman.