There are two issues with open fires, one is damage to the ground and the other is the risk of fire from sparks.
Where the concern is damage to the ground, as on campsites that ban open fires, I have been given permission to use my firebox (a yukon) sat in a tray, on a low aluminium table, once its explained how my "open fire" isn't going to damage the ground. In one instance I emailed a picture of my Yukon "in action" before being given permission.
Very few people are aware of the firebox (or hobo stove) concept and always associate a "camp fire" a large pile of flaming logs placed directly on the ground.
But in circumstances where the concern is fire risk, then the no open fire rule will apply to fireboxes, such my Yukon or your woodland edge, just as it would with an open fire. Even though, sparks will be more contained than with a fully open fire.
So it is safer to initially consider fire boxes as "open fires" until you have cleared it with the site owner.
Graham