My simple intuition is that if you assume a bivalent logic system, otherwise called two-valued logic system, it means that every proposition is either true or not true, and there is no "middle" ground, so to speak, or a third option, what you called "the excluded third."
Principle of bivalence
As far as I can tell Aristotle (3rd century BC) first wrote about this. In a slightly different way he writes about "law of non-contradiction", basically meaning that something cannot be and not be at the same time.
And it will not be possible to be and not to be the same thing, except
in virtue of an ambiguity, just as if one whom we call "man", and
others were to call "not-man"; but the point in question is not this,
whether the same thing can at the same time be and not be a man in
name, but whether it can be in fact.
Aristotle and excluded middle
This idea, that something cannot both be and not be is expressed in logical notation as ¬(P ∧ ¬P), said "not (P AND NOT P)". Basically something cannot be and not be at the same time. If this is true, and we a bivalent logic where every proposition is either true or false, then ¬(P ∧ ¬P) leads to (P ∨ ¬P), meaning something has to either be or not be.
Aristotle specifically refers to the "intermediate". He says 'intermediate among two contradictories."
He then proposes that "there cannot be an intermediate between
contradictories, but of one subject we must either affirm or deny any
one predicate" (Book IV, CH 7, p. 531). In the context of Aristotle's
traditional logic, this is a remarkably precise statement of the law
of excluded middle, P ∨ ¬P.
Aristotle and law of excluded middle
So as to why it's called the excluded middle, it may be because Aristotle called it the disallowed "intermediate between contradictories", I'm not sure about this. The term makes intuitive sense to me, if "true" is on one side, and "false" is on the other, neither true nor false would fall in between, the middle, and it's excluded, presuming you accept a two-value logic system.
The article goes on to Leibniz:
"Every judgment is either true or false"
And Bertrand Russell:
- Law of excluded middle: "Everything must either be or not be."
Also, if I may make a correction, Wikipedia doesn't mention that the original expression is principium tertii exclusi, it says it's "also known as":
The law is also known as the law (or principle) of the excluded third,
in Latin principium tertii exclusi.
Law of excluded middle Wikipedia article
It was originally written, or translated, I should say, from Aristotle, being "intermediate between contradictories".