I think that the answer that @chocolate linked is pretty spot-on when it comes to kanji.
The simple answer is that kanji don't really have names. When there are many thousands of kanji used in daily writing, that would also require that there are many thousands of names for the characters. These names would be in addition to multiple readings, and the meanings (some kanji have multiple meanings).
When a particular kanji is referenced in conversation that the listener does not recognize, the speaker will typically do the following to clear up the ambiguity:
- Choose a simple everyday word that is commonly understood. 教師, for example.
- Specify which character in the word being used. 教師｛きょうし｝の教｛きょう｝。
Though I am sure you are aware, grammatically, the の in this structure can be considered as the possessive particle. I include this detail for learners with less Japanese experience.
If the process I just described is insufficient in specifying which kanji is being talked about, another common practice is for the speaker to trace the kanji out with their index finger on the palm of their hand. Depending on the speaker, and how much understanding of the Japanese language you demonstrate, you may have them trace the character out on their hand as they say 教師の教.
A final (and ultimately the most complex) way to distinguish kanji is by explaining radicals used. I have the least experience with this situation, but you will see it commonly used when you start talking about specific fields of study, like medicine, science and engineering. If the listener does not recognize the kanji via the method described above, the speaker may resort to specifying which radicals are used, and occasionally where they are placed. This is extremely effective when coupled with tracing the kanji on the palm of one's hand. Though this method may not be effective at conveying the reading of the kanji, the meaning as derived from the radicals is usually more clear.
You will also notice that hiragana and katakana characters are similar in that they do not specifically have names like roman characters have. If we want to specify a character, し, in conversation, you would probably say something like ひらがな（の）し, though in some circumstances, the の in this case may be optional.
Answer to Updated Question
You are digging for something that doesn't exist. Just as is with the case in English, there is no 'master list' of words that are considered common. Such a list would be excessively long, and nearly impossible to memorize for these circumstances.
Instead, why don't you use the words you learn when you first learn the kanji? When learning new characters, the standard practice is to learn the most basic (i.e. common) words that use the character. That's the closest you are going to get to this 'master list' you are searching for.