After decades of hunting and thousands of trail camera pictures, I have concluded a deer could care less if it is raining or not. I have just as many pictures of deer moving in the rain as I do on clear days. I believe the myth that deer don't travel in rainy weather is a result of two factors:
1. Hunter do not like to be out in wet, cold weather so their system one in their brain creates a plausible story that deer don't like it either. This gives them plausible reason to stay in the warm, cozy house on those miserable days. If they do go hunting in these conditions once or twice and see nothing, this incorrect conclusion is reinforced and becomes a fact to them. (If your sample size is < 30, your conclusions do not have a high confidence of being factual.)
2. A much larger percentage of hunters hunt on dry days resulting in more deer seen and killed on those days thereby reinforcing the incorrect premise derived in number 1 above.
Correlation does not indicate causation.
Now I will say that my experience and trail camera data, does indicate that deer do not move as much during extremely windy conditions (winds >15-20 mph). My unscientific guess on the cause of this would be that they cant see and hear as well and feel less secure.