I've been granted a UK visa 3 times. My current visa expires on 29 December. I am planning to apply for a new 6-months visitor visa in January 2018. However, my passport expires in October 2018.
Do I have to have at least 3 or 6 month passport validity from the requested visa expiry date?
No you do not need "at least 3 or 6 month passport validity from the requested visa expiry date".
In the case of a 6-month visitor visa, the gov.uk advises "Your passport must be valid for the whole of your stay in the UK."
No further mention of validity beyond your stay (note that this is different from your visa validity) is mentioned explicitly.
Further search on the US Department of State website, assuming you trust the US government's word on this, is also "Passport Validity: Must be valid for duration of your stay (six months remaining validity recommended)".
