There are two well-documented computing devices that can lay claim to being the earliest examples of purpose-built Internet routers - The Stanford "Blue Box" router and the "Fuzzball" router.
The "Blue Box" is associated with Cisco Systems because the two founders of Cisco were Stanford employees, and used it as the basis for their first commercial product. It was originally developed by William Yeager as software for the PDP/11 to provide multi-protocol routing amongst networks on the Stanford Campus. Development began in 1981, but by 1985 the "Blue Box" router was running dedicated software developed by Yeager on dedicated Motorola 68000 CPU boards with 256KB of RAM also developed at Stanford by Andy Bechtolsheim, a future cofounder of Sun Microsystems. Also by 1985, Stanford was one of the major nodes on the growing ARPANET/NSFNet/Internet. While not particularly designated as "core" routing nodes, the "Blue Box" routers at Stanford likely handled as much traffic at the time as a core router, since they were gateways for the biggest edge node.
The "Fuzzball" router was also developed in 1985, and according to the U.S. National Science Foundation was the first true router for core usage. It was developed by David Mills under a grant from NSF and was used in backbone routing for NSFNet. Like with the "Blue Box", it was a specialty OS (which has the namesake "Fuzzball") purpose-built for routing and running on DEC LSI-11 hardware.
While both these early computers can satisfy being called the first dedicated router and a core router, it is mostly true that the "Blue Box" of 1985 was more of a dedicated router, having purpose-built hardware and software, and the "Fuzzball" was more of a core router, since it was in that role for the NSFNet, even though it ran on more generic hardware than the "Blue Box".
It should also be noted that these were very early examples, and pre-date the competitive push among companies like Cisco to make truly high-speed routers. Naturally, these later routers continued to use purpose-built OS software, but added ASIC and custom peripheral hardware to speed packet processing. Of course, Cisco Systems was a leader in this regard, and their device lineage clearly goes back to the Stanford "Blue Box".