I might be misunderstanding, but it seems like the landlord's income would include economic rent as well as profit by this definition.

You might imagine a residential lease where you agree to perform maintenance and upkeep and pay property taxes in return for a reduction in the lease cost. But this wouldn't reduce your lease cost to $0 - you're still paying for the privilege of being allowed to live there, so doesn't this fall under the traditional definition of "economic rent"?