the loin: the thick, fatty flesh low on the back, beside the kidneys
This is a body word before it is anything else. It names the soft place low on the back where the flesh is thickest, right next to the kidneys. Priests and butchers knew the spot exactly, because it is where the hard fat lies that Israel's law reserved for the altar.
Five of the thirteen uses are the same sentence, repeated as a standing instruction in Leviticus: take the two kidneys, and the fat that is on them, which is by the loins. That wording is a recipe rather than a picture. It tells whoever is cutting up the animal precisely which piece of fat to lift out.
Away from the altar, the same spot shows up on people. Job describes a rich man who has grown a roll of fat over his flank, a picture of soft and careless prosperity. A psalmist says his loins are filled with burning, which is what an ache in the small of the back feels like from the inside. The ordinary Hebrew word for the waist (מׇתְנַיִם) covers the whole midsection and is where a belt goes; this word points at a smaller, meatier target.