I will offer a third approach to composting weeds. I have a large bin (probably over 2 cubic yards), into which weeds, occasional grass clippings, and mulched leaves go. Nothing diseased, and no tomato haulms. It would be classified as a cold bin. The sides have large holes, as it is made from plastic lattice. I keep adding to this, and really only turn it when the bin is nearly full. That is about every 2 years. When it seems "done" the bin gets turned onto the soil a final time and cures. In the bin or on the ground, it is very biologically active (many worms, soil insects, isopods, millipedes, and so forth).
A new bin is started and the cycle starts over again.
The cured compost is mostly used as bottom material for new raised beds or for containers. Top material in the beds is usually municipal compost.
I never seem to have problems with weeds sprouting in that compost. I think you could consider this cold compost/long duration.