It does seem that the construction is relevant. The animal needs to see the fence and recognise it as such if it is going to be detered or attempt to get over. Secure tops or flexible tops make a difference as most animals won't land on a moving edge. The shape is also important as the animal will position itself to get over or onto the top of the fence. These points are general for animals e.g. a horse will run into a wire fence and be injured but will not run into a wall and foxes tend to use the top of the fence/wall to get an extra push with their hind legs. Having said that chooks have their own behaviours and quickly learn. Mine learned that they can climb netting and that a four foot stone wall is not only an easy ascent, but allows them to avoid other fences and walk into the neighbour's garden where they steal the dog food (! - if they keep on they will be the dog food!). Clipping wings might work, but again mine learned that they can fly like crabs to surprising heights.
Given that preamble, I'd suggest that for asthetics and to get sunshine into the children's area, you have two choices:
1. Use high mesh fencing. Our main run now has 2400mm high fences which are panels of mesh stapled to CLS timber and bolted together so that they can be moved. The advantage is that they are adaptable, reasonably cheap, strong and almost invisible (stain the wood). With a triangle support they have stood up to really ferocious gales and snow. The disadvantage is that they collect leaves.
2. Use a solid fence which would be a windbreak for the garden. A four foot fence with a high skirt sloping upward from the fence made of loose mesh would work without being too expensive (would look like a letter K without the bottom r/h leg). We've used this method successfully in our own garden and it can be used on gates.
HTH