In the paper we consider selected formal models, coming from the field of 'pure' mathematics, as well as from some related areas, in which the notion 'efficiency' appears. The presented essay may be seen as a continuation, development and (at the same time) specification of same ideas discussed in the previous article of the author: On manysideness, relativity and complexity of the 'efficiency' (as a category) (in Polish: O wielostronnosci, relatywizmie i zlozonosci kategorii efektywnosci). In addition to the proposals formulated in the above cited paper (concerning the classification and explanation of various 'kinds' of efficiency) we introduce some new ways of meaning of this term, which we suggest to call: (a) basis-type efficiency, (b) sup (inf)-type efficiency, we also define and shortly discuss the following three types of efficiency, related to partial (pre)orders and formal logics, (c) informative capacity (reflecting the 'richness' of an information contained in formulas defining given order), (d) linear similarity - efficiency (expressing a 'distance of the (pre)order from the linear part' of the order in mind), (e) logical efficiency. In the final part we put together (and compare) 'official' terms denoting 'efficiency' and related notions presently functioning in economics, management and praxeology. The further forms of the meaning of notion 'efficiency' are discussed in the 'twin' paper submitted for publication in the present issue of Mathematical Economics.