Replacing Vintage Carlon Thyrocon Rotary Dimmer with Lutron Skylark S603-PGH Sliding Dimmer

Happy new year! I’m trying to replace an old rotary dimmer with a newer slider/switch and could really use your help. I searched and searched and couldn’t really find someone with the same wiring conundrum I am about to explain.
It’s currently configured in a 3-way configuration. There is a rotary switch/dimmer (press in to turn on/off) that controls an overhead light in the dining room. The kitchen next door has a 2-gang with one switch turning the dining room light on/off. Dim level is persistent with the rotary switch – that is to say if you turn the light on from the kitchen, the dim level is that to which the rotary is currently set. On the rotary it says ‘Carlon Thyrocon’ and elsewhere on it 500W. I couldn’t find any datasheets or anything about this device besides that it may be ‘vintage’ based on a couple lot descriptions on ebay. It has 2 red wires and a black wire coming out the back. Here’s where things get confusing for me: the switch box seems to have a single 4-wire romex coming through – so red, white, black and bare copper ground. From the rotary to the romex it is wired black <-> black, one red <-> red, other red <-> white. bare ground is connected to the metal of the switch box. This is not what i was expecting for a 3-way configuration. As stated i have a lutron dimmer i would like to connect. Wiring available on the Lutron is one green, one red, one red with white stripe, one black wire. The Lutron instructions recommend connecting – again from switch to wiring box – black <-> tagged (i believe common) black, red <-> other black (i believe one traveler wire), red with white stripe <-> red (i believe other traveler wire), green to bare copper. So as you can see, the wiring in the box does not seem to match what is expected in convention. My guess is they just used whatever spool of wire to do the job so there is likely (hopefully) a convention in use throughout this house – but unfortunately no Rosetta Stone to translate between that and expected wiring convention. I could also not know what I’m talking about of course. I have attached a picture – i hope it helps to illustrate what i am trying to convey. Unfortunately i don’t have access to any test tools here (dmm for example) so i’m going to need a little ingenuity to get this thing hooked up safely. Thank you sincerely to anyone who’s reading right now (or made it this far) – I genuinely appreciate any advice you all may have.

• Jayce