@pokerfaceblonde Yup- not that much different, but more fridge space and sounds like fewer dishes for you.
You can definitely have a cooling jar and a collection jar. The CDC recommends not pooling warm and cool milk, so that would be in keeping with their recommendations. My understanding is that this is based more on hypothetical risk than documented cases of illness. Some people choose not to pre-cool and just dump it all together.
I measured the temp of my jar before and after adding warm milk and felt reassured it was ok- but definitely that is a personal risk assessment I made, that you might not feel comfortable with.
ETA: the pumping subs are awash with opinions on this, as you might imagine.