I'm the anti-Wittgenstein. In my view, it's ALL private language, or at least it might as well be. Only individuals know just how words correspond to the meanings they assign, and there's no way to share that with other people. Meanings are inherently private. (Meanings being different than definitions on my view.)
A couple specific comments about things brought up in the article:
One problem that arises from the private nature of this definition is that it is impossible to tell whether one has remembered the connection correctly. Whatever seems to be right will be right. There is no difference between believing one is right and actually being right about the connection [because you are the deciding judge], and thus a mistake in the application of the private word is impossible
At time T1 you might believe that you're associating term/symbol/etc. "#" with private sensation "@" just as you've always done, whereas it's not actually true that you've always associated "#" with private sensation "@," at time T2, you might recall this--you might remember, "Wait a minute, I was associating '#' with '!' instead--I made a mistake at time T2." So mistakes are not impossible. They just hinge on memory.
And that only means that here we can’t talk about right.
You can get right or wrong that you previously associated a meaning with a term, but re meaning in general, there aren't right or wrong meanings in the first place. There are common or conventional meanings versus uncommon or uncoventional ones, but it's not wrong to be uncommon or unconventional. Consensus usage certainly doesn't determine correctness. That would be an argumentum ad populum.
What Wittgenstein is saying is that the word ‘beetle’ cannot be referring to the beetle itself, because if it did then only I could know what I meant by the word ‘beetle’, as only I know what is in my box. In the same way, we can see that the word ‘pain’ cannot refer directly to the sensation, because only I could know what that sensation is: if the word did refer to the sensation, the word would mean nothing to anyone but me (as a word in a private language would).
And that is indeed the case--meaning is subjective, it is inherently private. That's a signficant part of how language works. There are public aspects to language obviously, such as the fact that I'm typing these words and you can see the text on your screen, but those pubilc aspect to not literally include meaning. Meaning is only had by you assigning whatever meanings you do mentally, and those can't be shared. You can only share marks on a screen, sounds with your mouth, etc.