Sometimes, it is desirable to identify a particular train rather than just finding any train that travels to a given destination. For instance, discounted train tickets in Germany are usually valid only on a particular train.
Train station display various information on scheduled trains, and my questions refers to that information. I would like to identify the information relevant for my particular train on such displays. This question is not about how to recognize the train when it is already waiting at the platform (as it might be too late then and I like to wait on the platform before the train arrives).
When travelling by train in e.g. Germany and France, information on trains is displayed along with the unique (per day) number of the train (cf. e.g. this display from Germany, or this one from France).
Now, I might be able to figure out which one is my train (even though it still seems risky to me, if an earlier train to the same destination happens to be delayed) if I know its departure time and its destination. However, especially for the destination, this is unlikely - unless I happen to ride that train all the way until its final destination, neither am I interested in where the train travels to after I got out, nor do I usually even know (at least German online tickets for trains do not indicate the final destination of the booked trains, I would have to specifically retrieve that information from the online itinerary system).
Practical example: When taking an IC train from Mannheim (blue marker in the map below) to Mainz (green marker), depending on the exact time of travel, the final destinations of these trains (red/orange markers) are Düsseldorf, Greifswald, Hamburg-Altona, Magdeburg, Münster, Dortmund, Cologne, and Mainz itself. While many of these are at least roughly on route when extrapolating the path from Mannheim to Mainz, it is, for instance, geographically rather counterintuitive to travel toward Magdeburg or Greifswald (orange markers) when one wants to go to Mainz:
So, even if you know where the final destination displayed for a train is, it is by no means guaranteed you could, without any further info, use that information to determine which train is yours.
Therefore, my primary question is: How are travellers "officially" supposed to identify one's train on such displays in Austria and Switzerland; what's the idea on how to use the information displays there when the train number is not visible?
A secondary question, which is kind-of contained in the above (only if someone happens to know; I think it would improve the answer), is why train numbers are not displayed as well in Austria and Switzerland, as internally, surely they must exist (obviously so for e.g. publicly numbered German trains that cross the border).
*Interestingly, the source page of that Austrian photo also shows various displays from Austria that do show train numbers. So, in Austria, the issue I observed only seems to apply to some stations.
EDIT: While Swiss and Austrian railways may not offer any tickets bound to specific trains, German railways sells such tickets for trips to and through Austria and/or Switzerland. As such, it is usually necessary at some point to identify a particular German train in a Swiss or Austrian station.
EDIT2: Another use case would be travelling in a group where some people board a train at a later time than the others. In such a case, they would need to make sure they are referring to the exactly same train, not just a train around a certain time towards a given place.