A handle on a bucket is called a bail.
Neither, they move up and down rather like a bucket handle. When the handle is "up" as in when you are carrying the bucket there is a big space under the curve of the handle. When the bucket is down and the handle is lying on the side of the bucket there is no space under the handle - that is how your ribs work.
The squeeze part of the mop bucket is just called a handle bucket. to find a replacement you would have to call your mop buckets manufacturer and ask if you could obtain a replacement bucket.
You call it a French Bucket or your face
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A bucket is a rigid container, often with a handle, used to carry liquid or small items, or the contents of this container.
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The definition of a bucket is an open container or deep cylindrical vessel that has a flat bottom, with a handle that is used to hold and carry liquids. A bucket can be made of plastic, metal, or wood.
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A light-colored bucket would be called a pale pail.
A light-colored bucket would be called a pale pail.