Hi, 'n welcome. It is confusing, isn't it?
Freud publish in FAQ on their website, cutting speeds for very large router bits as fast as 3000rpm.
Many manufacturers however recommend 400rpm (hard wood) and 500rpm (soft wood) for Forstner bits. I use a chart from Australian Woodsmith, laminated by the drill press and it says 400 and 500rpm.
I'm thinking that router bits cut small chips and, for half their rotation, actually fly through fresh cooling air. (Ignoring the blind fence breakout solution.)
Forstener bits, on the other hand, produce shavings and do their work surrounded by wood and sawdust/shavings. Also, F-bits rely on their outer surface riding in the hole to keep them straight. Thats how they work.
F-bits get hot in a drill press at 500rpm, no way I'd spin one in a router.
But I'd be pleased to hear other opinions.