Its just not logical to argue, either they are or they arent.
For instance, corporations can be bought or sold, but people cannot per the 13th amendment.
Help me understand how these inconsistent principles are allowed in the supposedly rigorous logic of the legal system
The proper reference isn't the dictionary. US socialization stems largely from the US Constitution. Within that framework, Person has a different meaning from the dictionary or most of the US legal frameworks. From that perspective, the objection to Person being ascribed to non-persons is obvious and warranted.
The US Supreme Court decided in 1886 [1] that it's the 14th amendment.
The general article on Wikipedia [2] has more info about it, and discusses the fact that corporate personhood is an abstraction that represents the rights of the individuals owning or running the company. "Statutes violating their prohibitions in dealing with corporations must necessarily infringe upon the rights of natural persons" and modern cases. That article also discusses how, from the 1920s to the 80s, general corporate personhood wasn't as broad as it is today. It also mentions, at the top, historical instances of the idea.
But to your point, no corporation in the US has full, equal rights to a natural person. It's an abstraction that the legal system does not apply blindly. You could change the phrase "corporate person" to something like "corporate legal entity with a set of rights that overlaps with natural persons" or demand a different approach to the rights of a corporation, but I don't think "you're using that word wrong" will hold much weight with legal professionals.
[1] https://en.wikipedia.org/wiki/Santa_Clara_County_v._Southern....
[2] https://en.wikipedia.org/wiki/Corporate_personhood#In_the_Un...
"Person" in legalese means something specific. It's not the same as the dictionary definition.