"...there are other definitions or stipulations concerning what "is"
The verb is is a tense form of the main (infinitive) verb be. So, if you study the different definitions for be, you would be studying is.
The Present Tense uses the verb is. The attributes of the Present Tense is that the present tense is used to state facts, show habitual action, but it is not used to state present action. Thus, is is a linking verb by itself, which is otherwise known as a "state-of-being" verb--intransitive, taking no object.
Take is and use it as as a helping verb (a.k.a. auxiliary verb) in verb phrase with a main verb, and you now can get all kinds of meaning, even action, as in the Present Progressive: She is going to college.
It's not a matter of is meaning equal, when you understand that, again used alone, it links one idea with or without a subject complement [predicate adjective/predicate nominative] to another.
The book is on the table. [adverb phrase telling where the book is]
That pen's ink is red. [the predicate adjective red describing the ink as being red]
She is an opera singer. [the predicate nominative singer modified by the adjective [nouns can be adjectives] opera; identifying or explaining that she is an opera singer]
Your statement, "The second is that "is" can also mean "the subject of this sentence currently manifests/embodies the abstract concept about to be listed" (e.g. He is sad." would follow this definition.)" is on cue with is functioning as a state-of-being verb with the purpose of being used as a linking verb:
He is sad.
Subject | linking verb | predicate adjective that answers a question an adjective would ask, "What kind?" What kind of feeling or emotion is he experiencing? That of being sad.
is alone means nothing. It's just a verb with no subject, no other words to help explain its purpose. I say purpose because after studying grammar for so long I began to see that what my textbook on grammar had been telling all the long, indirectly I might add, I had to literally put two an two together, to come up with this:
Every word in a sentence not only functions as a part of speech but also serves a purpose in the sentence.
And is is no exception. It's function is: primarily, present 3d singular of be--Webster's, and its purpose depends on how it is used: linking verb or helping verb. These rules give meaning to the word is. They define is.
Finally, our entire English language is base on a foundation: the eight parts of speech. Some laugh at this as being antiquated, irrelavant to today's linguistic/semantic advancements in grammar. There's only one problem with that, Webster's and all of these guys still use the eight parts of speech http://onelook.com/. And the last I heard is that is is still one of the eight parts of speech, a verb.