If I receive written permission to use content from a paper without citing, is it plagiarism?

It doesn't really matter that the author gave you "unrestricted rights to use any portion of their article" without the need of citing it. We, the readers, want to know that you're using it.

Because I have all the rights to do this, would this be considered plagiarism?

Yes.

Yes, it is plagiarism. The permission is irrelevant.

Consider this simplified but actually equivalent example: if your neighbor allows you to copy her answers in a written exam, is that not cheating just because she allowed you to use her answers? Of course not, that is ridiculous! It is cheating simply because you didn't come up with the answers yourself; whether or not you had your neighbor's permission is completely irrelevant.

You are confusing two orthogonal things: plagiarism and copyright violation.

Plagiarism is an ethical concept; it refers to passing off other's work as your own. Copyright violation is a legal concept; it means using a creative work in a way that is reserved for the copyright holder without acquiring a license from the copyright holder.

Those concepts are orthogonal: if you copy large portions from a book and you properly cite and attribute them, it's still a copyright violation even though it's not plagiarism. And in your example, it's still plagiarism even though it's not a copyright violation.

Note: plagiarism may come up in a legal context as well. For example, in most institutions of higher education, there are rules forbidding plagiarism, if you are a student, you often have to sign or acknowledge an honor code forbidding plagiarism, if you are faculty, a tutor, a TA, an employee, there might be a statute about plagiarism in your employment contract, making it a finable or fireable offense. Also, your thesis will have a signed statement that you did not plagiarize, and if you did, signing that statement may be considered fraudulent. In that case, plagiarism may not only have ethical/moral dimensions but a legal one as well (you get expelled / fired / fined / your degree revoked). In the case that the degree is a prerequisite for practicing in a legally protected profession (e.g. lawyer, doctor), falsely signing that document may even be considered a criminal offense or even a felony.

Plagiarism is passing of the work of someone else as your own. So regardless of the authors permission, this is absolutely plagiarism.

This is quite similar an increasingly common form of plagiarism where students purchase an essay or pay someone to complete their assignment for them. At the universities I've been involved in this is treated very seriously and students fail entire units when they are caught. As serious academic misconduct they can in fact be removed from a course entirely.