I mean, pullups and running are very different…

Running you just need anything warm, nothing else matters. Pullups (outside in the winter?) you can’t sacrifice grip and should go with something like Mechanix gloves. Just enough to get a layer between you and the bar.

So really, the solution is two items.

If you really want one single solution, military surplus FROG gloves.

They won’t be as warm as mittens when running, but you’ll be able to easily get your sleeves over them to seal up and trap air. And they’re designed to deal with wet/cold as well as the fire resistance they’re named after.

But still won’t have the straight grip as Mechanix gloves, or the warmth of bulky gloves/mittens that would be useless for pullups.