If we rule out Gen 1 to make things more interesting, which console do you think had the most influence on those that followed, and why?

Atari 2600.

The NES. It set so many rules and standards, both on the hardware design and game design side. It changed the economic landscape of console gaming, altered the dynamic between console manufacturers and third-party developers, birthed multiple legendary long-lasting franchises, and established a foothold for Nintendo that has lasted 40-plus years.

I can’t imagine modern video gaming without it.

Atari 2600/VCS slightly above NES/Famicom which is slightly above PS1. Atari 2600 is more primitive and less loved than NES and PS1, but its significance particularly in North America cannot be overstated.
2600 was not the first video game console or the first cartridge-based console. But it was the first cultural phenomenon and paved the way for others.

The PS1. Coming from a 16 bit SNES it was like entering a whole new world. It had the most diverse and creative waves of every genre. Gaming wasn't a children's hobby anymore. It changed the gaming industry into a mainstream phenomenon

Nes and Atari2600

NES and PS1.

In what way?

The NES. It set so many standards from the controller to game design to the host of IPs, first and third party, that were born on it. The library was just so far ahead of anything else on the market for its time.

PS1, the NES was important but gaming took off with 3D, 2D became just a side part of it, both are important and the PS1 stablished both things (3D > 2D, but should have both).

Also, the Dualshock is still the configuration of every controller out there, every single controller is a Dualshock configured by a different company.
Grips/handles on both sides, analog sticks on both sides, 4 shoulder buttons, 4 face buttons (SNES), dpad (NES), built-in rumble, pretty much every controller out there since it is built around it.

The PS1 defined the way people on consoles play up until this day.

Also it was responsible for creating a 3-axis power were America, Europe and Asia were all very important at the same time, making console sales pop the last bubble and become mainstream worldwide, for all ages.

There is nothing remotely comparable to the PS1 out there, the NES helped gaming get back on its feet in the 80s, the PS1 made it be what it is up until this day, for over 30 years now.