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"?
Rather, the phrase rent-seeking refers to seeking money without actually providing any economic value. It would be more accurate to view such payments as taxes (often levied by a private entity). Adam Smith categorizes income into profit, wages, and rent. Rent-seeking refers to the last one of these.
In a traditional residential lease, the landlord is responsible for the maintenance and upkeep of the property along with property taxes and, sometimes, some of the utility payments. Residential rent is based upon the value of the property as a residence, along with maintenance, taxes, etc that the landlord is responsible for. The income the landlord receive, according to Adam Smith's classification, is profit rather than rent.
Capital owners typically receive profit by controlling profitable assets such as stocks, bonds, and real estate.