Why is arginine's positive side chain classified as basic and not acidic? Arginine has a positive side chain.

Doesn't this mean it wants to lose a proton H+? Isn't something that wants to lose or donate a proton an acid?

Yet my book states, "arginine has a positively charged side chain, i.e it is
basic" .
Your textbook is imprecise. We consider whether a sidechain is acidic or basic based on the action of the sidechain when it is neutral.

Because arginine's sidechain is basic, it preferentially picks up a proton, and hence becomes positively charged. (In the same vein, carboxylic acid sidechains are acidic because they preferentially lose a proton, and exist as the negatively charged carboxylate anion.)