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Yes that works just fine unless they speak only Russian, or Chinese (which accounts for quite a large number of the humans in the world). The fact that you TLDR for a single paragraph means you have no interest in learning anything and are willing to stay ignorant. Good luck with that.
Just try to implement dd-MMM-yyyy in many places only to have the [non-technical] "stakeholder" go 'Why is the date f!@#$% up? Go fix that. Our stupid users won't unnerstand'. Yes Biff - going Biff....
Well, it always helps to have empathy towards the end-user in order to achieve as much ergonomic and intuitive programs, but back to your case, the programmer in charge of redacting the database transaction layer is supposed to make sure that the date is stored using a time-stamp, whereas the UI programmer should take care of accordingly parsing this time-stamp on an end-user localization basis so that the end-user can for example always remotely generate a consistent receipt whatever his location.
But yes, both programmers can definitely be the same person. Presenting information to the world in various notations or languages require more UI efforts than using more standardized schemes, but have its charms.
In all cases, if the data is presented ambiguously, then the project is an epic failure (!?), with data dumped to the end-user becoming inconsistent so unusable, but in theory and back to your initial question, inside a project with respective road-maps for several programmers, it would be the UI programmer's job to properly dump / format data that have been consistently stored.
Developers have a responsibility to understand the system that they are building, and preferably before they start building the system IMHO. How the system is intended to be used (via a User Interface), and by whom, is part of that understanding. Therefore the answer has to be 'Yes'.
Wow, I was surprised that this is even a question but pleased so many believe they are responsible for a user-friendly UI.
Absolutely, even if the programmer is given specific specifications, they need to validate what the user experience will be. We do not program stuff just to move data around. Ultimately a user must interface with the information and when they do, it should be as intuitive as possible. Nothing is more crazy making for the user than a UI that is confusing or ambiguous when the designer and programmer (and the whole team) have the power to make clean sensible user experiences.