But that's subjective. Even my grandmother in the early-80s said differently. Everyone's computer-time and our smartphones and our digital clocks we've plugged into the wall, all have switched over to Wednesday at 00:00:00 midnight.

If you go to a bar they will tell you, "last call is at two in the morning." If someone heard a noise they say "I woke up at three in the morning to look around." My mother called them "the wee hours".

Sure, colloquially, 01:20 is very late on Tuesday night, because the Sun hasn't returned to announce Wednesday, and the cock hasn't crowed. So that is another way of looking at it. Time is analog, the orbits of Earth and Moon are analog, and we're cramming them into digital approximations and binary categories.

But you've misidentified the problem. The problem is that different humans interpret differently. The interpretation is influenced by culture and language. My scientific culture says that 01:20 is on Wednesday morning, and that is in agreement with my dearly departed grandmother, my smartphone, my calendars on Google and Outlook, and honestly there is no outlier in my life, except you, and Crunchyroll(?) who would insist that "01:20 is actually 25:20 on Tuesday night", and that writing it in this way would "help" solve a problem, because we're just explaining how damned confusing that is to rational Americans or users of either 12- or 24-hour clocks.

So "the problem" in my book is that a hypothetical human looked at our hypothetical "01:20 Wednesday" as rendered by a hypothetical computer, and that human wasn't thinking like a computer, or an American, or that human was Japanese, and that human erred with their interpretation of time notation. Now I don't have any idea of what time notation "01:20 Wednesday" would prompt a hypothetical trip to the airport 24 hours late, so it is not like this human was applying a different but correct interpretation to this problem.

Why don't we write "very early on Wednesday morning" or "before sunrise on Wednesday" or "just after midnight" (the latter is literally what "a.m." means...) and it won't look like another typo. Because if you write "1:20am on Wednesday is Tuesday night" then I'm going to call out a contradiction.

Computers can actually help with this, because if I've made a hypothetical calendar entry for my flight and I email the .ICS file to you, then the timezone and day-of-week is baked-in to that, so if everyone's settings are correct, bypass human interpretations. Many airlines have a way to email your itinerary directly to your friends. Just use the tools.