Ring™ and Swann™ floodlight cameras
Couple of ways

1. Get the right product

I gather you are looking at this one. That's the wrong one, you want this other one, called the "Ring Spotlight Cam Mount".   Black and white wires go to supply hot and neutral, then the red wire goes onward to your external lights.  just as this FAQ explains.  

That one appears to have onboard lights as well (just not googly-eye ones), but I bet you can disable them in software. 

2.  It's a smart, networked product: Use it.

You have a smart-home hub that the camera talks to.  When the camera says "hey, I saw something and turned on", add scripting to that event that turns on the additional lights.  

3. Brute-force it

Presumably, this gadget draws more power when the lights are on.  Inside the box where you are hooking to its #18 wire, insert some of your own #18.  In which the "hot" makes about 30 turns around a magnetic reed switch, and the neutral does the same in the opposite direction.  There'll be some design considerations in sizing this thing, based on how much current it pulls, the ampere-turn rating of the reed switch, and the efficacy of your electromagnet building skills.   Get it right, and when the gadget pulls the higher amperage to light the light, that closes the reed switch, which operates the relay.  You can take it from there.