It seems to me that the underlying issue is that The Railway has failed to publicise a good name for what Trainline calls "Open Return", i.e. a return with unspecified return itinerary, and additionally historically used "Open Return" to mean specifically (anytime) period return, excluding day returns in particular.
Note that Trainline is not alone in including Day Returns under "Open Return". For example, the
GWR journey planner (which I understand is MixingDeck which is not powered by Trainline) has a tick box for "Open Return" and then offers both Day and Period Returns inside. Even worse, under "Amend Journey" the tick box says "Open Return (return anytime)" (see attached screenshot) and then offers Off-Peak tickets! Of course, this isn't really a problem, because when the passenger selects a specific ticket the name is displayed clearly, and includes "Day Return" or "(Super) Off-Peak" where needed. Nevertheless, things could be made clearer by splitting the initial "Open Return" option into "Open Return (day)" and "Open Return (month)", which will have minimal impact on passengers as most people know which one they want.
I haven't used Trainline before, but from trying some searches in the app, I think the big problem is that when they split tickets, the prominently displayed ticket name becomes "SplitSave", and the condition "Return same day" is shown in slightly smaller font below, which could be missed (see attached screenshot). An easy fix would be to extend the name to summarise the most restrictive properties of the tickets in the combination, e.g. "SplitSave (Advance)", "SplitSave (Anytime Day Single)", "SplitSave (Super Off-Peak Return)" (ignoring the fact that some Super Off-Peak restrictions are less restrictive than some Off-Peak restrictions).
The issue of the SplitSave name hiding the day/period return distinction is compounded by the fact that the overall cheapest ticket is shown at the top, separately from the two clearly grouped "Return same day" and "Return within 1 month" options below. Since, for day/period returns, "Return same day" could never be more expensive than "Return within 1 month", showing the cheapest ticket at the top could be abolished and the tickets would still be displayed in the same order (with the cheapest ticket now being at the top under the "Return same day" heading, if a day return is available).
The other thing to add, is that this is happening after people are specifying overnight stays. They are specifying to return on different days to their departure
Are you certain that this happened? I.e. a passenger specified two separate dates for out and return, and got offered a ticket (or a split involving a ticket) that is only valid to return on the same day? Are you able to reproduce this in the app now? (I couldn't figure out how to see what the individual tickets in a split are without buying them, but one could infer from the ticket price if it's a simple split into day/period returns, since the split point is displayed.)
a lady who had what Trainline were describing as an "Open Return" but was infact an off peak day return and two advance singles routed via a split at Crewe, but because she thought she had a return to Nottingham valid on any train for a calendar month she boarded the direct service.
Surely Trainline could not have sold Advance Singles if the passenger ticked the "Open Return" box and thus did not input a return journey? More likely the passenger selected an out and return itinerary, and was offered a SplitSave (with the small text below saying "Specified train(s) only"), but somehow misunderstood/misremembered?