>Sounds pretty similar to a calculator with a numerical root-finding algorithm, if you only substitute board position it has never encounterd with a polynomial it has never encountered.

>What does it mean for a deterministic algorithm to make decisions?

Newton-Raphson isn't choosing among possible actions. Given (x_n), its next step is mechanically specified by the update rule: compute the derivative, take the tangent intercept, repeat. The intermediate result changes the next input, but that's not by-itself decision making.

A chess engine, again, does something different. From a position, there are many legal actions it could take. It considers alternatives, estimates their consequences according to some objective, and selects one. The engine has to work out which available move best advances its objective.

You can make both algorithms determinstic, but determinism isn't the distinction i'm drawing. 'Decision' here doesn't mean some metaphysical excercise of free will. It's more about selecting an action from alternatives based on an evaluation of their expected consequences. Determinism is orthogonal to decision making. Deterministic doesn't mean predictable, nor does it make its choices any less it own computation.