Is there a specific term to refer to a word associated with specific part of speech?

I'm looking for a term I can use to refer to words tagged with a specific part of speech. Let's say the word run may be used as a verb and as a noun. So, run (n) and run (v) are ... of word run. Is there a term I can use to fill in the gap?

In looking at several sources, they all use the phrase parts of speech, so that seems to be a common term.

After that, more of them than not use the single word function. But both category and class are also used.

Note, too, that while most sources list eight parts of speech, and give common names to them, others will use a different number—and sometimes different names.

So, you're not going to find any universal authoritative source on terminology (or even classification). When writing about parts of speech, you're going to have to simply choose your own terminology and remain consistent.

According to Practical Grammar: A Canadian Writer's Resource, by Maxine Ruvinsky:

Every word in a sentence can be categorized as one of eight parts of speech depending on its function—what it does—in that sentence. The following are the eight parts of speech:

She goes on to mention nouns, pronouns, verbs, adjectives, adverbs, prepositions, conjunctions, and interjections.

Part of Speech says:

In the English language, words can be considered as the smallest elements that have distinctive meanings. Based on their use and functions, words are categorized into several types or parts of speech. This article will offer definitions and examples for the 8 major parts of speech in English grammar: noun, pronoun, verb, adverb, adjective, conjunction, preposition, and interjection.

Butte College says:

There are eight parts of speech in the English language: noun, pronoun, verb, adjective, adverb, preposition, conjunction, and interjection. The part of speech indicates how the word functions in meaning as well as grammatically within the sentence. An individual word can function as more than one part of speech when used in different circumstances. Understanding parts of speech is essential for determining the correct definition of a word when using the dictionary.

ThoughtCo says:

A part of speech is a term used in traditional grammar for one of the nine main categories into which words are classified according to their functions in sentences, such as nouns or verbs. Also known as word classes, these are the building blocks of grammar.

It further says:

The parts of speech are commonly divided into open classes (nouns, verbs, adjectives, and adverbs) and closed classes (pronouns, prepositions, conjunctions, articles/determiners, and interjections). Although we can add to the open classes of words as language develops, those in the closed classes are pretty much set in stone.

Note the addition of articles/determiners as a ninth part of speech.

Oxford Dictionaries says:

All words belong to categories called word classes (or parts of speech) according to the part they play in a sentence. The main word classes in English are listed below.

It lists noun, verb, adjective, adverb, pronoun, preposition, conjunction, determiner, and exclamation. Like ThoughtCo, this includes a ninth part of speech (determiner)—and Oxford renames interjection to exclamation.