How do you read the date 13 June?
Is it thirteen of June, thirteenth of June, thirteenth June. Do you say 'of' before the month? I think first day, after month is British but I wish other English speakers also answer.
The Cambridge Dictionary article below explains how writing and speaking dates may differ. This point is implicit in the article. Reading or saying this out loud directly (as written) leads to using the unnatural "thirteen June", which few native speakers would ever say. Basically, you can write "13 June" or "June 13", but you wouldn't read or speak it that way. You would say "thirteenth".
Sometimes the last two letters of the number as spoken can be used (th, rd, st, nd):
Today is the 7th September.
The grand opening is on 1st June. or … on June 1st.