It's not always necessary to choose a fully enclosed litter box for your cat. Studies have shown that cats don't really care whether the litter box has a lid or not. A fully enclosed litter box can be convenient for owners because it prevents litter from splashing out and also controls odors. Some cats value privacy and prefer to use a secluded litter box, while others find a fully enclosed litter box oppressive and prefer a spacious, well-lit one. Therefore, it's best to choose a litter box based on your cat's preferences. 
Types of cat litter boxes
1. Open litter box. This is a common type of litter box that can meet the basic needs of cats and does not make them feel restricted. However, it also deprives cats of privacy and makes them more likely to dig litter outside the box.
2. Semi-enclosed litter box. It is slightly higher than an open litter box, which can act as a barrier to prevent the cat from digging the litter outside.
3. Enclosed litter box. These usually have an entrance/exit, either at the top or side. They are suitable for shy cats who don't like being disturbed while using the litter box, and effectively prevent litter from being carried out and odors from spreading. However, don't choose one that's too small, as the cat won't have enough room to move around.
4. Single and double-layer litter boxes. Single-layer litter boxes are suitable for litter types that clump when wet, such as mineral litter, paper litter, clumping wood litter, and tofu litter. These require frequent cleaning to prevent bacterial growth. Double-layer litter boxes are suitable for types that do not clump when wet, such as wood litter and crystal litter. The upper layer contains litter, and the lower layer contains an absorbent pad, preventing waste from accumulating in the box and causing odor and bacteria.
5. Automatic litter boxes. These come in fully automatic and semi-automatic versions. Semi-automatic boxes require manual cleaning of the litter, while fully automatic boxes can be cleaned automatically at set times. They are more convenient, but more expensive and require the use of special litter.