Excluding Gen. 1...
I understand "influencing" to mean the act of establishing "industry standards"... Abandoned standards are of lesser relevance; those that persist are of greater influence.
Two systems seem like obvious choices to me.
PS1, Why:
Controller:
It evolves the basic design established by NES/SNES, being the largest individual contributor to modern controller design, with contributions such as:
1 - Responsible for popularizing the use of analog thumbsticks with potentiometer technology, a standard used throughout the entire industry even today.
2 - First to use pair of analog thumbsticks, the standard of all modern controllers
3 - First to use L3 and R3 buttons on the analog thumbsticks.
4 - first to Add of 2 extra buttons on the shoulders. (L2-R2, ZL-ZR, etc.)
5 - First to add internal motors in the controller grips for vibration.
6 - Ripping off Evolving the design established by the SNES. It set the standard for modern controller ergonomics, allowing for agile transitions between Analog Thumbsticks/D-pad/Face Buttons.
The controller is so absolutely influential that its configuration (established in 1997) would be able to play most games on the market in 2026. The "standards" it established continue to be universally followed throughout the industry.
It popularized media with the exclusive function of storage:
Many tried to use CDs, all failed. Playstation 1 sold over 100 million units and established digital media as the industry standard. Digital media has become so utterly dominant that many cite it as a key factor in the original Playstation crushing the N64.
Media and entertainment functions:
A modest beginning, the Playstation served only to listen to music. But this is where the idea that consoles can serve other purposes besides gaming began.
Xbox 360, Why:
1- XboxLive:
For years, Xbox Live set the "standard to be followed among consoles." The entire industry followed their example, undeniably. DLCs, digital game purchases, subscription services... it all started here.
2- The first modern console:
User interface gained a central position on consoles, apps, game lists, etc. all of this began with the 360, standards maintained to this day.
Playstation OG was the "Cd music boy" - Xbox was the "Netflix and other apps" Man.
3- The controller:
1 - Triggers, shoulder buttons were a thing of the past! The trigger was indispensable for playing Halo, Battlefield, Call of Duty, etc.
2 - "Xbox" button - The button that took us to the UI, later copied by the entire industry.
3 - Wire is a thing of the past. It was the first to bring wireless control as a standard.
4 - Made for adult hands. (This is a personal opinion, fell free to ignore it, but... I feel that before the Xbox 360 controllers were designed for small hands.)
Now here is my take...
If your main focus is OS, Network and Services as part of the console > Xbox 360 is the most influential, it shaped modern consoles.
If your main focus is hardware > The original PlayStation is the console that set the most standards in terms of hardware, in especial, the controller and the disc drive that popularized the use of Digital Media.
