Invoking cylindrical Bravais lattices, Adler (1974, 1977) proposed a mathematically precise definition for the botanical classification of phyllotaxis. It is based on opposed pairs of parastichy families, that are conspicuous and visible. Jean (1988) generalized this concept to non-opposed pairs of parastichy families. In the present paper it is shown that this generalization implies a notion of conspicuity different from Adler's. This is made obvious by redefining the key terms of the two approaches. Both classifications are well defined. For Adler's, this is shown by presenting a general proof for his conjecture that conspicuous (in the sense of Adler) opposed pairs of parastichy families are visible. There are indications that in applications to models of phyllotaxis (van Iterson model, inhibitor models) their solutions are better characterized by Jean's classification. The differences between Adler's and Jean's classification show up only in very rare cases, so that the practice of pattern determination is only insignificantly touched by the present results. It turns out that the widely used contact parastichy method to determine phyllotactic patterns gives results according to Jean's classification rather than Adler's.