> Independent Judiciary means independent of the executive and legislative branches. It doesn't mean immunity to public rebuke.

This seems to be confirmed by the first sentence of the wikipedia article on "Judicial independence", but the second sentence also mentions not being subject to influence from "private or partisan interests".

>Plenty of judges have to renew their mandate by standing for election.

this is mostly a US thing, and only in certain states. Federal judges are appointed AFAIK.