What to use to separate the garden?

Then take the green fence. Visually unobtrusive and a boundary for the dog.

Then take the green fence. Visually inconspicuous and a boundary for the dog.

I think my current tendency is rather towards the picket fence from I think it suits my style better visually and fits the garden.

One type of willow fence would also lean more towards a natural style and can be designed in different ways. One of our future neighbors has one because of his dog.

Double-rod mesh fencing is basically the new chain-link fence. Everywhere, sterile, boring. Best of all lined with plastic sheets. Yuck.
I would say it doesn’t fit your garden at all.
Choose something more natural. Picket fences would be an option.
I also understand the desire to separate it. Especially with children and dogs.
Maybe with a gate(arch) and from there slabs as a path to the beds. You want to get to them with the wheelbarrow after all. A compost bin at the back. Bound gravel would be an alternative, but just don’t step on it after heavy rain, then it gets muddy.

Double wire mesh fencing is basically the new chain-link fence. Everywhere, sterile, boring. Ideally lined with plastic sheets. Ugh. I would say it doesn’t fit your garden at all. Choose something more natural. Picket fences would be one option. I also understand the desire to separate it off. Especially with children and a dog. Maybe with a (gate) arch and from there slabs as a walkway to the beds. You want to be able to get there with the wheelbarrow. A compost bin at the back. A bound surface would be an alternative, but don’t step on it after heavy rain, it gets muddy.

We understand each other I also find the picket fence good. A gate arch that is overgrown is also a great idea. The compost is in the back corner and slabs are a good idea. Thank you.

Don't think in limitations, but in sight lines. What can you see from where, which new perspective opens up for you when you go around a "corner." For structuring, you can also use waterways or a pond. Terraces or similar areas, possibly with a pergola and climbing plants, are also suitable. A garden shed or a greenhouse can help with structuring. Espalier fruit trees are also an elegant way to separate visually. There are really countless possibilities; all the above fulfill the aspect of "low maintenance."

With the dog, it depends on which preferences it develops: running away, digging, marking, "full-speed racing"... after that, choose the enclosure, because the dog is supposed to stay there and maybe not ruin vegetable beds.