No. Interest is the portion of economic surplus that goes to Capital, whereas Rent is the portion going to Land (with Wages going to Labor.) These are the classical Factors of Production going back to Adam Smith, David Ricardo, and the very beginnings of modern economic theory.
The neoclassical redefinition of economic rent is the source of the confusion here. Historically, that redefinition happened primarily to discredit the views of economist and politician Henry George (in the late 1800s) but we've been stuck with the terribly confusing definition ever since.
Here is an excellent paper by Mason Gaffney that details the historical motivations: Neo-classical Economics as a Stratagem
against Henry George
There is a more recent notion that "rent" is defined in terms of its properties: being unearned, being the result of a lack of competition, etc. All of these are derivative, secondary aspects. Rent comes from exclusive control — typically granted by government — over some resource. Historically, land was the canonical example of such exclusive control. But nowadays, we have not just real property but "intellectual property" and monopoly rights granted in the form of exclusive use of portions of the electromagnetic spectrum, rights of way granted for the laying of utility cables, exclusive rights to operate (e.g.) taxi service within a specific area, etc. These are all forms of "land" in a more general sense, and all of them generate economic rent in virtue of those monopoly rights. Clearly such exclusive control also results in a lack of competition, unearned income, etc. But again, those are merely consequences of the monopoly rights; they do not define rent. A lack of competition may simply be due to market inefficiency, in which case it is not rent. It's just inefficiency. Rent only arises when one party is granted monopoly rights over a resource.
Once you recognize that all rent comes from control of a resource granted through monopoly rights, distinguishing between rent, interest, and wages becomes very simple. It is only the confused definitions that make it difficult.