I push the lid *mostly* on. I actually have a single bucket with a fermentation lock from beer brewing, but I don't use it - as I said, the benefits don't match the effort for me. It will absolutely generate some CO2 as the process goes on.
and with a good stir to start, so things are well...
The confusion around (2) - the middle method - is that its been mixed up with "wet mash". I often feed my birds wet mash, which isn't fermented (though can be) its simply crumble stirred with water until it absorbs the liquid, then served as an oatmeal like paste. Helps control waste, seems to...