Earth cycle 245 is one of the big problems in the modern world. When does it change over to 246? Do I wake up on the 245th but the date changes before lunch? What happens with billing systems, calendars, and what not?

Left/right, port/starboard, north/south are all in common use because we need to be able to talk about directions relative to different frames of reference. It’s just as true that we need to talk about time in multiple different frames of reference.