I just realised that the meaning of the term "metabelian", when applied to groups, or Lie algebras, seems to have changed over years. (These days, it means that $[[G,G],[G,G]]$ is trivial, while in the past it was occasionally used to indicate that $[[G,G],G]$ is trivial. The difference here is that between solvability and nilpotence, that is.)
This wiki says "The concept and term metabelian group was introduced by Furtwangler in 1930. The term metabelian was earlier used for groups of nilpotence class 2, but is no longer used in that sense." (I don't understand "earlier" here. Can that sentence be parsed uniquely? Earlier than Furtwangler introduced the term? Earlier than the wiki article was written?)
I know at least one reference from mid 1960s (a PhD thesis from the US) where that old-fashioned usage is present, and I'd like to understand the history better in this instance. Thanks for help.
EDIT: it appears that the situation with usage and its history may be even different for groups and Lie algebras; I only dealt with literature on metabelian Lie algebras, and from answers so far I gather that there may be difference.