Language modeling and its consequences have been a disaster for the human race. In fact, we shouldn't have named it like this at all; we should have called it something more appropriate like "word kebab generator" or "word sequence model". This mislabeling has created misunderstanding and misinformation. To understand the distinction, we are going to dive into the specifics of "machine learning" and the game of kings.
The truth about machine learning
A machine-learned model could be represented most easily as a large statistics machine. We feed it a ton of sentences and the machine can analyze them: it can analyze which words are used, the context of those words, the position of the words in the sentence, and how common these words get used with other words. Add a little pseudo-random magic to make less common words and sentence structures rarer and violà you have created a machine that knows enough about words to make a sentence. This next example will serve us well in the near future: you could feed a model every chess position that has ever been played. Provided you set up the right context (such as board state), you create a viable chess computer!
Is it AI?
...however, it is not intelligent. There is no intelligence present in this chess program; all of the intelligence has been fed to it by an actual intelligence - the thousands of chess players providing the moves for the data model. This model then simply superglues moves together and picks the move most relevant to the current game situation. There is no thinking. We have not created an intelligence, we have only created a program to navigate the complex data of a million chess games.
To reiterate, the distinction is clear: this model is not thinking. We have not created intelligence; we have only written a program to extract data out of intelligence. So, how do we go about building an intelligent chess computer? Logically speaking, that must mean it thinks for itself. It should require no training data, as this would be supplying the intelligence to our machine, and not creating it. If we write a program that understands the rules of chess, is able to judge positions, and think ahead, we would create an artificial intelligence. We have created a program which solves a problem (and thus acts intelligently) without having supplied it with our own intelligence.
The sweet spot
Finally, imagine if we were to combine our knowledge of machine learning and artificial intelligence. Previously, we had to write the "board judge" ourselves; however, if we build a machine learned model out of chess positions for this, we can apply our artificial intelligence to data that our human intelligence created. In fact, this is already how the best chess models work. By combining AI and ML, we have created something more powerful than we thought was possible.
I hope this clears up some confusion. If you read this all the way through, you have earned the right to slap the next person to mention some "speech recognition AI" or the next "DALL-E AI Image Generator".