Thank you!

Both are very nice. I would pick the woven. I would chose it for two reasons.

The woven texture may remind you not to over load it with items. safer to take down

The second reason. The wood box is more solid then the woven. It will lesson the impact damage of accidentally dropped (Being filled with less items and the material itself)

anyone you chose, enjoy your new apartment.

Pick the woven one, if they have you want to place it length wise, look for a smaller size one, otherwise, otherwise, they may slide off when you open cabinet door. You might like to do what I do with my baskets of towels in bathroom, add a large keyring holder ( I find really large ones here) and tie them to the basket through the metal ring with a bright ribbon or just some string like rope, both look really well. I suggest you also cover what you have in them with one of those disposable plastic table covers or even better a tea towel to protect from dust. It will save you a lot of work anyway. With plastic table cloths you just throw them away and recover with a new one when cleaning, with tea towels, just wash them. They won't be seen either.

I chose chalkboard because although woven is prettier, I am concerned about the huge amounts of dust that we all know accumulates on the tops of kitchen cabinets. Ideally, I think the storage container would have a lid, and would allow for periodic wiping down or dusting.

There are other options. I've recently seen something by Rubbermaid called bento storage, which comes in a variety of colors and fabrics. Some have lids.

Neither. Build your own long box with shelving board and hinges so the front board is attached to the top of the box next to the ceiling and lifts up to access the space. Painted the with color of the ceiling, the box(es) would visually disappear.

If, when you move to your next home, the box you've built it doesn't fit your new home, you could always stand it on end and add shelves for a place in a clothes closet to hold shoes.