In everyday communication, the verb to implicate is typically used in locutions of the form ‘to implicate X in Y’, where X is some person and Y is some crime or something similarly disreputable. We can thus read, in a newsreport about some scandal, that newly discovered evidence implicates somebody, who hasn’t been suspected before, in the scandal. Using it in any other way, e.g. to say that Mary implicated that she did not want any coffee, would strike most ordinary speakers of English as odd. The verb to imply, on the other hand, has a much wider range of uses; it would thus be perfectly normal to say that Mary implied that she did not want any coffee. The two words, as used in everyday English, are thus clearly not synonyms.
The verb to implicate, and the corresponding noun implicature, are, however, sometimes also used as technical terms in linguistics and related fields. In that technical sense, it can be said that Mary implicated that she did not want any coffee by saying that it would keep her awake. What that means is that, by her words, Mary communicated that she did not want any coffee, even though she did not say so explicitly, nor did she say anything that strictly speaking entailed that. By logic alone, one cannot deduce ‘I don’t want any coffee’ from ‘coffee would keep me awake’, but Mary’s words nevertheless conveyed that she did not want any coffee, in virtue of certain generally understood conventions about how human communication proceeds.
In the theoretical contexts in which the word implicate is used in this way, the word imply is typically used in a narrower sense than in the everyday ones: in that narrower sense, X implies Y only if Y can be derived from X by logic alone (without invoking the conventions about how human communication proceeds). In that terminology, one would say that Mary’s words did not imply, but that she merely implicated, that she did not want any coffee. The words imply and implicate are thus not synonymous in this theoretical usage either.
When the quoted dictionary definition gives imply as a meaning of implicate, it is explaining the technical term implicate in terms of the everyday sense of imply.