According to my study,
A field could be "an open area of land without trees or buildings". In this case we say "they're playing in the field"
But, a field could also be "an area of land that is used for sports". In this case we say "they're playing on the field"
Do we say "the children are playing on/in the field" depending the kind of area of land that they are playing in/on?
I think that "on the field" and "in the field" are often used rather interchangeably in such contexts, with limited regard for what kind of field it is. If it is actually a totally undeveloped meadow, "on the field" seems less likely, but in informal speech might still be used.
"in" the field is much more common in my experience, but I wouldn't call a sports ground a "field". I would be more specific, for example, "They are playing on the football pitch".
As shown here in the field is more largely used. However, if the action of the verb concerns the field itself (I am just thinking of another context), then on the field is perfectly reasonable:
They built a hotel on the field next to the hippodrome.
