The word rent has a general meaning, and a specific technical meaning within economics.
As the article you've linked to states explicitly:
Rent-seeking occurs when an individual or business attempts to make money from its resources without using those resources to benefit to society or generate wealth
It should be obvious that when an apartment is rented out, money is being made by using the resource of the apartment to benefit society.
The rent (in the lay sense) on the apartment is not the same as an economic rent, and at that link, we read:
Economic rent is an excess payment made to or for a factor of production over the amount required by the property owner to proceed with the deal.
The word rent has two different meanings.
One is the lay meaning: the price paid for the use of property for a fixed period of time.
The specific technical meaning within the context of rent-seeking is as set out in the link in your question. This is a different meaning to the lay meaning. It is not the same meaning. It is different. The same word can be used to mean two different things. Even though it's the same word. One word, two different meanings. We use the context to determine which is the appropriate meaning. As the context in your question is economics (something of which you are well aware, as you have posted this question on the Economics SE), then the technical meaning, as defined in the link you provided, is the relevant one. Not the lay meaning.
So rent-seeking, is about seeking economic rent (technical sense); it is not about charging the rent (lay sense) for the use of the property.
Lobbying and rent-seeking are different things. Lobbying may or may not involve rent-seeking. Rent-seeking may or may not involve lobbying. It's like asking whether fish is the same thing as chips. You can have fish with or without chips. You can have chips with or without fish. Just because they often go together, does not mean they are the same thing.