I assume that saying that a word is not a part of standard English means that the word has not been recorded in the dictionaries (bar some cases of terms obtained through free production, or terms about which there is not a majority of dictionaries agreeing about them). So, "that's not a word" means "that's not in the lexicon".
("Irregardless" is a word of chiefly North American English, but it is labelled as non-standard or jocular in the SOED.)
For all words, there exists a more or less extensive period that you could call its "twilight zone", while it is not yet in the dictionaries, but has nevertheless acquired some currency. It may never survive this period and finally be forgotten, that is, never be recorded. During this time of incertitude, the word will exist for some people, those that will brave the sanction of the dictionary, and will act as if the word exists—there is often enough not much braving needed as there exists a crying need for a new term and a more or less satisfactory candidate to fill the blank—, and there is no arguing with those as to its existence; for them, the word is a part of the language. For other people it will not be considered as a word and those will revert to other words or do without it altogether, in good conscience of the idea that the concept does not deserve a word or that there are words that will name the concept more aptly. It follows that the idea contained in the second sentence of the OP is not realistic.
If people didn't understand it, I would think they would rather say, "what do you mean?", than making it a word after a definition is explained.
People who do not understand it will say "what do you mean", but then that is not an accurate description of the cause that can lead to the recording of a new term in a dictionary: they can ask what it means but that does not give anyone the possibility to "make the given form a word", that is, to give it a definition that will ensure that everyone from then on will learn the word according to this definition; there must first be a great enough number of persons asking for the meaning; if this is so the word has already some currency, which shows that the word is already found in the talk of some people and/or in their writing. But this is only a step in a progression towards greater and greater currency and there is no authority that is aware of all the steps in that progression; authorities that can determine whether the word has an impact that is important enough in the language must first become aware of one of those stages, and that particular stage has to show a degree of currency that is substantial enough, otherwise it might even be hardly detectable by these authorities. In modern linguistic systems (except Esperanto), words have to strive in order to come into being, and they will go on striving after that.
