While I am a strong supporter of ISO8601 (e.g. 2023-07-12) there is an argument that decreasing specificity is preferable (as in ISO8601) and that for most day-to-day usage the year is somewhat implicit, hence being at the end.
Personally, I write the month out if I'm not using 8601 formatting to avoid any ambiguity.
For any sort of recordkeeping, though, I think it's preferable to go full on YYYY-MM-DD. It's more thorough, precise, and sorts properly on a computer.
In most contexts I would never use the cardinal number "3", it'd always be "the 3rd of May" or "May (the) 3rd", but in a country like Australia (with a high number of speakers from different cultural backgrounds) you hear all sorts of conventions used in casual conversation.
Personally, I write the month out if I'm not using 8601 formatting to avoid any ambiguity.
For any sort of recordkeeping, though, I think it's preferable to go full on YYYY-MM-DD. It's more thorough, precise, and sorts properly on a computer.