While reading history, I've noticed some historians making a distinction between these three terms, but then not really explaining the differences in context. For example, in A Student's Guide to the Study of History, by John Lukacs, he says that the best works on medieval history are not found in the works of medieval chroniclers but in more recent books. The crux of my question is besides the point Lukacs made. What is the meaningful distinction, if any, between these three terms?
Historian and Chronicler are basically the same thing. Lots of medieval works of history are called the Chronicle of Such and Such or X's Chronicle, or sometime's just Chronicles (most famously but not exclusively the work of Jean Froissart). This gives rise to referring to the author as a chronicler since they wrote a chronicle, which makes sense. Modern historians that try and enforce a distinction between medieval chronicler and historian usually do so out of a belief that medieval authors were biased and unreliable and thus are a lesser student of history (a chronicler) as opposed to the modern objective historian. This is nonsense, some medieval historians were pretty bad at being historians but some were great and you could say the same thing about modern historians. In the end there isn't really a meaningful distinction between the terms.
Annalist is technically something different. An annalist is someone who wrote or maintained an Annal. An Annal was a record of major events that was updated annually. These were often maintained by monasteries or cathedrals and they would usually only have a sentence or two for the events of a year. They were also, although not always, very focused on local events with only major international moments breaking through that. There's no attempt to construct a broader narrative over these years and it's generally assumed they were updated as they happened rather than being written years later with the benefit of hindsight. That said, there are literally hundreds of surviving medieval annals from all over Europe and there will always be exceptions and deviations from the standard pattern.
It's also worth noting that even these lines sometimes get blurred. The Anglo-Saxon Chronicle, one of the most valuable works on pre-conquest England, is actually a series of related but somewhat independent annals maintained over centuries and not actually a chronicle at all. It was named by modern historians who clearly were not respecting the distinction between a chronicle and an annal. Of course those same historians invented the idea of an Anglo-Saxon race so there were already some red flags there..