Both uses of this notion can be found, and in many cases the distinction is simply ignored, see for instance examples for the first and second meaning. In informal discussions the distinction is seldom important, and tendency is towards the broader second meaning. When studying the structure of polynomials however, one often definitely needs a notion with the first meaning. This is for instance the case when considering a monomial basis of a polynomial ring, or a monomial ordering of that basis. An argument in favor of the first meaning is also that no obvious other notion is available to designate these values (the term power product is in use, in particular when "monomial" is used with the first meaning, but it does not make the absence of constants clear either), while the notion term of a polynomial unambiguously coincides with the second meaning of monomial.