"Rent" in the colloquial sense of payments pursuant to a residential lease is a different meaning from "rent" in the economics sense of the word. The use of this word falls into the trap of many words with differing colloquial and technical meanings. I don't see how abandoning a word's technical definition because of a competing colloquial definition helps anyone.
The problem is that rent seeking, in the colloquial sense, is an attempt to paint a normal activity, like getting paid rent for housing, with the same brush as bribery type behavior.
It's very similar to say, how Trump co-opted fake news to just mean any news he disagreed with. Original creators of the term fake news are rightfully mad because technically it should refer to news that is clearly factually untrue, like pizzagate.