playing football 'ON the field', 'AT the field' or 'IN the field?

Based on my references, it's actually 'on the field'. (British English)

E.g.: The children are playing football on the field. (for a playing field)
BUT; The cows are grazing in the field. (for a field with crops)


*The preposition 'in' is also used for 'playground', 'park' and 'garden'.
*The preposition 'on' is also used for 'beach' and 'farm'.
*The preposition 'at' is also used for 'market', 'school', 'hospital' and 'airport'.

However, when you refer to a meeting point, you can use 'at', for example; "Meet me at the school field."