The "given" is "Whatever is immediately present to the mind......?

I agree that this is a standard usage of the term. There are of course different and competing ways of understanding "the given" -- that is, of understanding the idea of immediate presence. One of the big quandaries involves the notion of a "present moment" when the given is supposedly being experienced. Does the given include our most immediate memories, or are they already not-so-given after all? If it doesn't include our immediate memories, how can there be room for anything being given within an infinitesimal thing like the present?

I don't believe anything is given to the mind. It has to work for everything it 'gets', and in this sense is not separate from the perceptual apparatus. If there is a moment or 'stage' at which something mental has not been elaborated, I would simply call that moment, a part of the "perception" and leave it at that. There is no "staging area belonging to objects", before we think.