Free and unfenced sports fields are a disappearing and valuable asset and should be clearly visible so that people can find them. The gated commercial versions should be distingusihed by being coloured red isntead of being a bland green. I will start by adding notes like “open” or “gated” to fields I come across on walking around and urge others to do the same, but Iassume that a general change in colouring would be far more efficient
I tried in vain to launch a design proposal, but the form was inaccessible omn my computer (presumably ciuse by the safety features), so I thought a post on this forum might succeed in getting field accessible to anyone coulored as such rather than mixed with the opposite kind.
As to your comments on orthogonality, the key propery of a free field is that anyone may enter without owning, paying or being hindered in any other manner.Too many areas are labelled as sports fields where for instance children wishing to play ball. In coutries where open access to nature is specified by law, ownership of the land is irrelevant.
I understand your concern and agree that the accessibility of a sport field to the public is a valuable information. The same applies to playgrounds for children.
Anyhow you have to be aware that mappers like you and me and others here in the forum do not have any influence on the rendering of certain features in any of the countless applications and maps using OSM data, including the standard map at https://www.openstreetmap.org. If you have any rendering proposal you would have to direct it to the designer of the map or app, as @Mateusz_Konieczny pointed out.
For the purpose of indicating the accessibility there is the well established key access. Sport fields and playground which are free accessible to the public can be marked with access=yes, those which are restricted in use can be marked with access=private or access=customers.
By “sports field” do you mean sports pitches or something different?
Sports pitches may be free and public but still have some controls on access, e.g. they may be fenced and closed at night or there may be a booking system.
In any case for a distinction to be rendered it would first need to be tagged on the OSM data, and it’s not clear if what you have in mind corresponds to existing tags. Pointing to example OSM locations might help.