I take mine out and lay them on the side, but that is just personal preference. You'll find some people on this forum that think leaving them pointy side down in egg cartons or such is the only way to go, and others that adamantly think laying them on the side is the right way to do it. Both ways work.
My reasoning for taking the turner out is partly to make clean-up easier. They can make a mess when they hatch. Why leave the turner in there? It is just something else you have to clean. If there are electronics or mechanisms involved, that could get a bit touchy with water.
The other issue is how many sharp tight corners are there that little legs could get trapped in? With my turner there are quite a few. I don't see a lot on yours.
On the clean-up issue. Do you have a layer of plastic or something under everything that can be removed and either easily cleaned or disposed of. Mine has a piece of plastic that has the water reservoirs in it that covers the bottom. It is easy to take out and clean after the hatch is over. Just something to think about. That shelf liner may work for that.
Good luck!!!