I think if gaming were separated into a BC and AD the birth of Jesus moment took place during the Playstation era.

From a technical aspect, you had the obvious shift to rendered polygons vs hand drawn sprites. More important, I think, was the rise of the "AAA game", which is what many gamers think of when they think of gaming these days. These big budget, action-oriented, and narrative-focused games played out like Hollywood summer blockbusters and featured things like (actual and/or good) voice acting that you didn't really see before and has become the norm since.

Games like that became possible because of a few things: a large enough user base to support these increased budgets, a mature enough user base to demand more in-depth experiences, and the technology to make them possible. I'd say the Playstation, as the first 100+ million selling console nailing the 18-35 demo, was the console where that all came together and paved the way for modern gaming as we know it.

Specifically, I feel like Metal Gear Solid (98) was the first game to tie all the elements of what we think of as modern video games together (but you could make a case for earlier stuff like Resident Evil (96)).

Now if the Playstation and Metal Gear are the birth of Jesus for the medium, you could certainly argue that the NES represents that "In the beginning/seven days" part of the Bible, and I don't think you would be wrong. At the same time, looking at where games are currently, they're typically more in line with Metal Gear than Mario Bros.