It seems this was a personal itch that Mozilla's developer Graydon Hoare wanted to scratch and then Mozilla saw potentional and started sponsoring it for Servo.
So Graydon Hoare made the initial design and implementation, but it was significantly redesigned and reimplemented after it was picked up by the wider Mozilla organisation for use in browser development.
So Rust as we know it today was definitely made for browser development at Mozilla. But that means both Firefox and Servo, not just Servo.
Afaik it existed on the prototype stage before Mozilla investigated using it because it fit their purpose. But I think it's fair to say we know about Rust today because of Servo.
