If you don't butcher it like you did, ISO8601 caters for any amount of precision you need.
For the vast, vast majority of my usage 2023-07-23 is sufficient. If you need a time as well just append the time and the nice thing is it'll still keep things in order (I've not found myself needing to use the timezone notation as well since I don't usually share dates cross-border). For work I use the week notation a lot 2023-W30-4.
04-06-13 is not helpful because now I don't know if you're European and mean 4th of June 2013, or if you're american and mean 6th of April 2013, or if you're some weirdo who means 13th of June 2004.
It's like when the crazy guy says it's easy and then pulls out a pinboard with pictures and string connecting them and proceeds to explain how it makes sense in his head and you have to admit that you sort of follow but also can't believe what you're hearing is reality.