If a lantern is mounted on a handle it can only be fueled by holding a bag of coal/wood next to it and using the thumbstick to transfer. the issue with this is that you cant drain that bag of fuel entirely. it can only go down to 2 so it can remain a pouch. And you cannot take a single piece of coal and just put it into that lantern. you first have to remove the lantern from the tool handle, place the coal in the lantern, and then put the lantern back on the tool handle.
not much of an issue if youre in town and fueling from a giant pile of coal. but if youre in the mine, running low on coal and you need to mine coal as you go to fuel your only source of light while surrounded by enemies you may not have time to pick up the coal, put it all in your pack so its in a pouch, get the pack off, get the coal pouch out, put the bag back on your back, get your tool with the lantern on it, thumbstick most of the pouch into the lantern, and then either store the last bit or work through disconnecting your lantern and going through that process. you should, for example, if youve got a lantern mounted on your pick be able to quickly mine coal with one hand while grabbing the coal and stuffing it into that lantern piece by piece as you go.