I'm a retired wildlife biologist. I recall (but wouldn't swear to it) that deer can't see the red light.
Night Wing, my buddy uses the tacks and ties along his trails and the red light works well to light them up. This friend will even take a rake to the woods after leaf fall to clear debris out of the trails he walks in on to help him move quickly (some of the walks in are nearly a mile)and quietly.
Where I hunt there are no headlights in the woods. Of course the deer are used to such lights when they are around roads and houses. However, just like they expect to smell people when they get around our homes, they don't expect it in their core areas. They probably don't expect bright lights in the "woods" either.
However, having written all this, I can't say I've ever spooked a deer with a light, even with white light. But the reason is because I wouldn't know if it was from my sound, my form, my odor, or my light that caused the spook?