Why do you conflate socialism and a lack of democracy? Am I wrong in thinking there can be a democratically elected socialist government?
Yes, if a socialist government is democratically elected it's overthrown shortly afterwards by the CIA.
(sarcasm, but this is basically what happened to South America, and there's longstanding questions about some of Italy's postwar politics)
If the largest democracy in history isn't a "notable example" what is?
https://en.m.wikipedia.org/wiki/Forty-second_Amendment_of_th...
Even the link you provided notes the same thing: "Ambedkar's (the original author of the constitution)...objection [to putting "socialist" in the original preamble] was [it was] "purely superfluous" and "unnecessary", as "socialist principles are already embodied in our Constitution" through Fundamental Rights and the Directive Principles of State Policy."