No, at least not in the sense that "hallucination" is used in the context of LLMs. It is specifically used to differentiate between the two cases you jumbled together: outputting correct information (as is represented in the training data) vs outputting "made-up" information.

A language model doesn't "try" anything, it does what it is trained to do - predict the next token, yes, but that is not hallucination, that is the training objective.

Also, though not widely used, there are other types of LLMs, e.g. diffusion-based ones, which actually do not use a next token prediction objective and rather iteratively predict parts of the text in multiple places at once (Llada is one such example). And, of course, these models also hallucinate a bunch if you let them.

Redefining a term to suit some straw man AI boogeyman hate only makes it harder to properly discuss these issues.