Was down at Aldi yesterday and the weatherproof jackets are a bit on the heavy side, but at £20 who can argue. I settled for one of the showerproof / windproof tops for a tenner.
Could you really run with zips fully done up and a hood to keep dry, if not the rain will get down your neck in no time. If you don't find it too warm a buff round the neck can soak up some of the rain and allow you to have the collar slightly more open. You will still end up wet on the inside from sweat, as we build it up quicker than breathables can get rid of it.
I favour my montane litespeed which I wear for all sorts of activities. waterproof-ish, windproof and weighs just 6oz. I wear it for walking, cycling, running and all sorts of other times. Best £50 quid I have spent on clothing for years.
I have sealskinz socks which I wear for cycling but they are a bit shapeless and I think I might get blisters if I ran in them. They are several years old so newer ones might be better shaped.
Just about to head out the door for my run and it's chucking it down. First thing I will do when I get back is have a shower - amazing what you look forward to when you are already wet ! Maybe if the rain was as warm as a shower we wouldn't complain.