I agree, most people consider plant Spines, Thorns, Prickles, Teeth, the same thing because they are sharp and all serve the same purpose. The following definition is worded almost exactly the same at a few sites; Wikipedia, Revolvy, etc.:
~ In plant morphology, thorns, spines, and prickles, and in general spinose structures (sometimes called spinose teeth or spinose apical processes), are hard, rigid extensions or modifications of leaves, roots, stems or buds with sharp, stiff ends, and generally serve the same function: physically deterring animals from eating the plant material. ~
Since Cacti spines are borne from areoles, they'd probably be placed in the same category as Thorns. I do see both areoles and spines in your photo but only a small amount of the stem. To my eyes, the appropriate gallery would be Thorns, with a notation in the caption area saying "Spines emerging from areoles."
