Replacing Vintage Carlon Thyrocon Rotary Dimmer with Lutron Skylark S603-PGH Sliding Dimmer
Colors don't mean much
The color of wires in house-wiring cable (NM, UF) is only mildly meaningful as to their function, as you have observed, due to the fact that all such cables use the same color scheme (black/white/bare, with red for a second hot, blue for a third hot, and red/white striped for a second neutral).  As a result, grouping and connectivity are the main pieces of information that convey the meaning of wires in household wiring.
This is easy, though
What you're looking at here is the end of a 3-way switch spur; the existing dimmer is wired with its black common wire to the black wire from the cable, its red traveler wires to the red and white wires from the cable, and its ground connected to the bare ground wire in the cable.  The new dimmer wires up much the same way as a result: its black wire goes to the black wire from the cable, its red and red/white (traveler) wires go to the red and white traveler wires from the cable, and its green ground wire connects to the bare ground wire from the cable and a grounding pigtail to the box.