DNA is a long molecule that is made of many individual smaller molecules (called nucleotids) that come in four variants (called A, T, G, C). So a DNA molecule is a sequence that can be represented as ATGCTGCCTA...

This is a sequence of characters in this representation, but it's also a sequence of something resembling characters in reality. The cell has a component called "ribosome" that can take this sequence of characters as input and uses it kinda like a blueprint, and produces a protein (enzyme) depending on the blueprint. That enzyme can have many varying functions. So yes, this is a complex system.

The flow of information goes mostly in one direction: that is, from the cell nucleus's DNA to mRNA (intermediary step) and then to the ribosomes, where proteins are produced. Still, many parts of this process resemble script and communication (the transport of information), which I call "language".

Maybe I should have said entities?

How would you define entity them? Why not have the state of fitness communicating with possible future states, or evolutionary processes communicating with themselves, like a massive comment thread unbroken over 3+ billion years?

Calling it dialects sounds more like anthropomorphizing it to make it sound more like a language

Anthropomorphization? Kinda, sure. But different brances of life do use DNA/RNA differently, like us with separate chromosomes and one-way genes, prokaryotes with circular chromosomes that can be read in both directions, and viruses with free genes to hijack other DNA/RNA systems. We don't understand all the ins and outs of genes yet, and I'm certainly not an expert in the topic, so maybe it would be more accurate to call these variations "language groups" and instead use the compatability of the proteome and metabolome to define languages, with genes simply being the words and some syntax.

it somehow doesn't feel like a language to me when the information is just past further down the line and there is no real back-and-forth?

different categories of language

This I can almost agree on. A technical manual or a journal is definitely a form of language, but you can't have a conversation with it. A category of languages that can be used for conversations would exclude most genetic languages, as well as all programming languages. Both could still be used to define a communication protocol though, which might still be considered a conversational language. For example neurotransmitters or HDMI. The depth of this communication could be quite shallow though, even if they carry greater meaning in a different format.

Going further than this we could define a catergory that has enough communication depth and mutability to directly transmit arbitrary ideas, which would pretty much cover only primate, corvid, and cetacean language. At this point we've moved beyond the details of grammar and vocabulary and into the methods and capabilities which generate and use grammar and vocabulary. Certainly an interesting category to define and explore the limits of, but definitely not what most people think of when you say "language". Perhaps "laguability" would be a better term for this?

In my opinion, there's definitely a huge amount of difference between our conversation and what bacteria do with and between themselves, but language probably isn't the right word to differentiate them. It's the difference between a die cast toy car, a pickup truck, an ICBM, and a novel; vastly different yet all called vehicles.

I think the advantage of thinking of DNA as some kind of program code is that we can draw inspiration about what can/can't be done from IT. And the other way around, nature's DNA code might give inspiration to computer language development.