What does the phrase "reducing atmosphere" mean in quantitative terms?
I’m an environmental chemist and we frequently refer to ‘oxidizing’ or ‘reducing’ conditions in the environment. I’ll use soils as an example. The ‘oxic zone’ is the area of soil that oxygen can reach by diffusion and is seen as being under ‘oxidizing’ conditions because the reduction of $\ce{O2}$ is the dominant redox reaction that occurs in that zone. Elements that have multiple oxidation states tend be in the higher oxidation states. For example, sulfur will exist primarily as sulfate ($\ce{SO4^2-}$). As you get further away from the Earth’s surface, the oxygen is depleted (by microbes) and the redox conditions change: the dominant reduction reaction changes. Usually in this order: $\ce{O2}$, nitrate, manganese(III,IV), iron(III), sulfate and finally $\ce{H2}$. When the conditions become so reducing that the microbes are using $\ce{H2}$ as an electron acceptor, they making methane via a process called methanogenesis. This order arises because of reduction potentials. Microbes get the most energy for the least effort from reducing $\ce{O2}$, so they do that until it runs out, which is when they switch to the next best thing available.

Ultimately what it means to be a ‘reducing environment’ is that if you added a compound with a higher reduction potential then the dominant redox pair, it would become reduced. For example, if you could put $\ce{SO4^2-}$ in a methanogenic environment, it would be reduced to $\ce{H2S}$ very quickly.

I’ve read that the composition of Earth’s atmosphere in pre-life earth was very different from today. If I recall correctly, there was no $\ce{O2}$ as it was formed by microbes later on. Without $\ce{O2}$ in Earth’s atmosphere, it makes sense that conditions would be very reducing based on an analogy to the soil redox zones.