Here is the thing about running in rain: you are going to get wet. A fully waterproof jacket does not prevent this, it simply relocates the water to the inside, where it arrives from you rather than the sky. Anyone who has finished a wet hour in a sealed shell knows the feeling.
Once you accept that, the problem becomes much easier. The goal is not to stay dry. It is to stay warm and be able to see.
A cap does more than a jacket
The single biggest improvement to running in rain is a brim. Rain in your eyes is what makes a wet run miserable — it is why you end up squinting, hunching and looking at the ground. A cap with a stiff enough brim keeps your face clear and your head up, and it weighs nothing. On a genuinely wet day it will do more for your mood than any other item you own.
Choose wind resistance over waterproofing
Being wet is tolerable. Being wet in wind is not, because moving air strips heat from damp fabric several times faster than still air. A light wind-resistant shell with a water-repellent finish will keep you more comfortable than a heavier fully waterproof one, and it will not turn into a sauna on the climbs. Save the real waterproof for walking, where you are not generating much heat.
Avoid anything that holds water
Cotton, heavy fleece, thick socks. Once saturated they stay saturated, add weight and chafe. Thin synthetic layers are the answer, even if that feels counterintuitive when it is cold. Two thin layers will also beat one thick one, because the air between them is what keeps you warm.
Afterwards
Get out of wet kit immediately — the cold arrives when you stop, not while you are running. Dry your shoes away from direct heat, and stuff them with newspaper rather than putting them on a radiator. Direct heat hardens midsole foam and shrinks the upper, which is how a good pair of shoes dies in one evening.
Wind-resistant layers sit in Jackets & Vests.