I share similar concerns with you here in the Philosophy Stack Exchange:
There is a great article on this in PsychologyToday where the author makes an attempt to draw some parallels between Dreams and Many Worlds Interpretation. Here it is: https://www.psychologytoday.com/us/blog/dream-catcher/201407/dreams-and-the-many-worlds-interpretation-quantum-physics
Another great source of information on this topic would be Eva T.H. Brann's work entitled: "The World of Imagination: Sum and Substance" which I highly recommend and would argue is the best source to begin your search for an answer.
Even the CIA has a take on it here: https://www.cia.gov/readingroom/document/cia-rdp96-00788r001700210016-5
A part of the problem is whether or not dreams are an experience of a "place" (Topos) that one goes to. Since you have experienced what a lucid dream is like then for you this would seem to be obvious; insofar as you would refer to it as an "alternate universe" (Or a 'simulated universe'). That is what it has seemed like to me in moments of Lucid Dreaming that I have personally experienced.
As to what the factors are that determine the level of resolution between a dream and the visualization of the chessboard as you put it is indeed difficult to ascertain. I cannot provide an answer in all honesty but the resources shared above should provide some food for thought.