The helpful information provided by the IRS is an attempt to translate the definitions under the law, so if you think you are in some third category that isn't classified as either, you consult 26 USC 7701. For example,
The term “person” shall be construed to mean and include an
individual, a trust, estate,partnership, association, company or
Companies and associations would seem to not be people: however, "corporation" is given a definition ("The term “corporation” includes associations, joint-stock companies, and insurance companies") so that associations are deemed to be corporations and therefore persons. The IRS restated the law to put those two parts together.
The law defines "domestic":
The term “domestic” when applied to a corporation or partnership means
created or organized in the United States or under the law of the
United States or of any State unless, in the case of a partnership,
the Secretary provides otherwise by regulations
The term “foreign” when applied to a corporation or partnership means
a corporation or partnership which is not domestic.
In case a particular configuration of the terms isn't explicitly covered by the definitions (for example "domestic individual") then the words have their ordinary meaning. But there are also specific definitions of "US person" in (30) and "foreign estate or trust" in (31). The terms used in a definition may be defined elsewhere, so you have to cross-check.
If there is a real uncertainty, this resource may help. It is not a logical paradox that a description does not cover everything that exists, for example my dog and the Russian embassy don't seem to be either domestic or foreign persons. But you should get advice from a paid tax attorney to determine if a situation that you have in mind does not fall under one of these two categories.