Here is a helpful excerpt from the lecture notes for Introduction to Molecular and Cellular Biology at Columbia University, taught by Lawrence Chasin and Deborah Mowshowitz (emphasis mine):
Re: noncompetitive vs. allosteric inhibition: noncompetitive inhibitors bind to a site other than the active site and render the enzyme ineffective. Allosteric inhibitors do the same thing. So, how are they different? And, in what way can we apply the Michaelis-Menton equation to our understanding of allosteric inhibitors? For instance, can we quantify what happens with the presence of an allosteric inhibitor, or do we just have a qualitative understanding?
I agree that at a simple mechanistic level non-competitive and allosteric inhibition appear the same. There are several differences, however. Allosteric inhibition generally acts by switching the enzyme between two alternative states, an active form and an inactive form. It usually works by binding to a sites in a specialized subunit of a mutimeric protein, and thus binds at several sites. The more inhibitor that binds, the more then can bind, and vice versa with substrate. The kinetics are thus complicated, being cooperative, and non-Michaelis Menton, and are beyond the scope of this course. So a qualitative understanding is all that is called for here. Allosteric inhibition is designed into the proteins and represents an important physiological process.
Noncompetitive inhibiton is more of a catch-all for non-physiological inhibition that does not compete with substrate for substrate binding to enzyme. In that, it is defined (and named) from a negative point of view. As described in your texts, a non-competive inhibitor may bind to a non-substrate site on a protein and distort it to the point of non-functionality, and adding more substrate will not alleviate this inhibition. Or, as in the example I used in lecture, it may simply block a catalytic site without interfering with substrate binding, an example that is more distinct from allosteric inhibition.
In addition, note that an allosteric inhibitor may display alternative kinetics in reference to substrate binding, as described in this NIH guide:
An allosteric inhibitor decreases activity by binding to an allosteric site, other than or in addition to the active site on the target. This interaction is characterized by a conformational change in the target enzyme that is required for inhibition. These conformational changes can affect the formation of the usual enzyme-substrate active site complex, stabilization of the transition state, or reduce the ability to lower the activation energy of catalysis. Figure 1e and Figure 2a are classical examples of allosteric inhibition. As such, an allosteric inhibitor may display a competitive, noncompetitive, or uncompetitive phenotype with respect to substrate binding.