The minimax polynomial minimizes the maximum error (the L-inf norm). The truncated Chebyshev series of an arbitrary function does not have that property (though it often comes close).

I thought the truncated Chebyshev series of an arbitrary function minimizes the L-inf error among all polynomials of the same degree, because Chebyshev polynomials have minimum L-inf norm among all monic polynomials of the same degree.

Let's try this: Tell me how you would construct a minimax approximation according to your definition?

The truncated Chebyshev series does not generally minimize the L-inf error among all polynomials of the same degree (unless the function being approximated is a polynomial of degree n+1). Constructing a minimax approximation is typically done via the Remez exchange algorithm, which is iterative, fussy, and prone to convergence failures. However, none of those matter when designing an offline approximation as is the case when you're writing math library functions.

Thank you, that was enlightening.