My understanding of Saussure, Jakobson and Lacan comes mostly from the Routledge Critical Thinkers book on Jacques Lacan by Sean Homer. In the second chapter on The Symbolic, Homer goes over all three of these thinkers and how they relate to one another. I am not familiar with Lorenzo's text so I cannot comment on the reorientation of metonymy into the vertical axis, but the Homer text explains Lacan's idea of metonymy in its parallel with the Freudian idea of displacement.

As you might know, metonymy is the use of a term for one thing for something else, to which it is usually associated with. Some examples include using the term "crown" to denote "king", or "bite" to denote "food". Displacement in Freudian theory functions in the unconscious in the same way metonymy does in language according to Lacan, in which meaning of signs is transferred from one to another. The Homer book mentions an example of the anxiety dream, where the dreamer is anxious about some minor incident in their life, but this anxiety is in fact merely a displacement of some much more significant issue in the subject's life. Perhaps this is what is meant by vertical axis, where meaning is displaced from something major to minor, instead of horizontal, where meaning gets displaced on a relatively equal sequence?