The usage is irrelevant if we're interested in cost per token. If you use it half as much, you get half as many tokens at half the cost. It's still $5.56 in electricity per million output tokens either way (using $0.20/kWh, adjust accordingly if you have cheaper electricity). If you use the API, you also pay half as much if you use half as much.
I think that's the biggest difference for most. If you can amortize the hardware costs, then 'burst usage' is cheaper at home to a degree, because you are paying a fixed monthly rate elsewise. Overall thought for most, it is likely cheaper to use the cloud than at home, but really depends on what you want.