This quote from the link you give surprised me somewhat, given that it could potentially involve four, four digit, numbers:
"The outside lamp mentioned earlier was used to light the engine numbers, which were recorded for every train, and signalmen developed expertise in remembering the numbers of two double-headed trains simultaneously."
Steam-era signalmen had to watch passing trains for tail lamp(s), and possible problems not visible to the train crew, such as hot boxes [grease boxes could catch fire] plus deal with sending 'train entering section', possibly while something else was happening on the other line. Dealing with all that and remembering, even briefly, four potentially four digit numbers, or trying to write them down in poor visibility, probably while those on the far line were possibly obscured by the train on the nearer line, or the weather, or drifting steam, seems a lot to ask, especially since human short-term memory for numbers, especially under stress, has its limitations.
Another factor is that, in the days of small locos each 'owned' by single crews and with limited coal capacity, long distance trains commonly changed engines during the trip, sometimes more than once, whilst locos sometimes had to be changed unexpectedly because of failures. Quite how this related to train reporting puzzles me. Local trains were presumably of no interest to control, and through passenger trains, especially expresses, were usually infrequent enough for them to be readily identifiable by timings, and were unlikely to get out of course anyway, so control would presumably only be interested in long distance freights, but even so...
Unlike the locos, the one constant in any long distance freight, barring failures, was the brake van. At about the same time as large numbers appeared on tenders, goods brake vans gained racks on the side for code letters indicating the class of train, its origin and destination, which are what control would be interested in. The racks seem to have gone out of use in the mid-1920s as early LMSR vans had them from new, but later ones did not. It's way outside my period of interest, but it seems more likely to me that the outside lamps were there to read the code on the van, not he loco number(s).