The U.S. Army still has the Army Doctrine Reference Publication (ADRP 1-02) which concerns the icons and symbols used to designate friendly, neutral and enemy forces. These are the standard terminology and visual aides used to provide a graphic representation of the battlefield, be it on a sand table or a PowerPoint presentation.
The specific symbols will vary based on the particular army and time period involved but every command and fighting unit in history has had a need to graphically represent the battlefield and troops (friendly or foe) involved.
War gaming is an important part of military planning. It is still part of standard operating procedure for the Military Decision Making Process used by the Army. It is a mandated step following course of action development, and when done correctly would resemble what your question described.
Historically, war games also hold a great deal of importance. The Department of Defense contracts with a number of think tanks to do this. A prominent example is the Center for Naval Analysis which has a department dedicated to conducted war games on behalf of the Department of the Navy. War games often serve the purpose of filling in for actual experience, particularly when actual conflict would occur on a scale too great to find historical or other experiential examples. CNA did frequent war games centered on World War III during the 1980s including both nuclear and non-nuclear scenarios.
War games can also inform policy development. Red Storm Rising by Tom Clancy and Larry Bond was loosely based on war games conducted by the CNA and highlighted the importance of Iceland in the ability of the U.S. to reinforce NATO during WWIII. Newt Gingrich, Jesse Helms, and others held the book up as a realistic depiction of warfare and used it as evidence to support favored programs during Congressional debate. Gingrich actually held a copy of the book aloft during a house debate in 1986 asking his fellow representatives if they had read the book, noting that anti-satellite technology "is what wins us the war."
A more modern example of the importance of war games and fiction in military planning is Ghost Fleet, which came out this year and simulate a war between the US and China/Russia.
TL/DR: War gaming is a critical part of military planning that substitutes for actual experience.
Edit: Here is an example of planning in World War II which demonstrates what OP is talking about
Are you referring to war games, where commanders would simulate a battle/war, or are you thinking more of an operations bunker/war room, where troop movements and observations would be updated in near real time and command decisions would be made during a battle? Your description seems to indicate the latter, and yes these were used, such as the RAF's command bunker in WWII. I am not too familiar with how widespread this sort of set-up was, but I'm fairly certain that this sort of large-scale overview of battles was somewhat common for commanders near the rear lines for some time.
I'm no historian and can't speak to the past, however I am somewhat knowledgeable with the present. What you are talking about is referred to as Common Operational Picture (or COP) in military terms, at least in the US. Using table maps is a bit archaic these days as there is a proliferation of technologies specifically designed to deliver real-time information to troop commanders.
Primarily these systems are used to give planners and coordinators the situational intelligence needed to make informed decisions, and one of the major advantages of a digital COP is that everyone who needs updated information gets that information at the same time. This is important because it reduces the chances of encountering friendly fire and provides for efficient distribution of resources.
Some of the more common tools in use by the US military today:
So what's really cool about these systems is that they also allow for simulation data, so that's where war gaming comes in. The military regularly conducts training exercises in which simulation data is fed into these systems. This data is usually scripted without the involvement of the commander or exercise personnel so that they have no prior knowledge of how the situation will unfold. It is supposed to give the most realistic effect possible to the simulation.
Additionally, there are other simulation systems that combine real personnel movements with virtual personnel movements in order to create larger scale exercises with fewer participants. So if your exercise is primarily for, say, reconnaissance training, you may have live personnel performing that function, and then have virtual personnel act on the intelligence. All the while, the information is fed into the COP systems as if everything were live. Sometimes the exercises are purely for command staff, so all field personnel are simulated.
So I don't know if I've helped at all, but hopefully this will be informative to someone, somewhere.
The Prussians came up with a game called kriegspiel (literally "war game") that used dice and pieces and was played on a large table broken up into grids that was used to train officers. I do not know if they also used the same idea (small representational pieces on a large board) to display real enemy positions during war though.
Western Approaches Command, which was the UK headquarters for the Battle of the Atlantic in WW2 is now a museum. They have the operations room on display as it was during the 40s complete with a big map on the wall and a map on a table with models representing boats and subs etc. http://www.imgur.com/gmcWILg.jpeg
What you seem to be thinking of is the terrain model phase of delivering an Operations Order. When going over the Scheme of Maneuver, you'll have a miniature layout of your area of operations, which can be anything from the detailed system described in your question at the battalion level and higher, to a 10x10 feet square of sand using sticks, rocks, or MRE trash at the squad level. Simply put, it's to orient everyone as to how they will actually move after stepping off, in a more effective manner than simply saying "Go northeast 2 miles, then southeast for 1 mile, then east for another mile," and so on. When you use a terrain model, your troops can actually visualize the route to the objective and instinctively know when to go where based on pacing or recognizing terrain features, as opposed to simply giving directions and distance where you have to write it down and designate a member of your lead element to navigate with a compass or a GPS.
