When should we use the word Stand and when Bay for ATC taxi clearance?
The correct terminology for a place where aircraft park is stand or aircraft stand, where bay or parking bay is colloquial.
The official definition from the ICAO ANNEX 14/I Aerodromes is:
Aircraft stand. A designated area on an apron intended to be used for parking an aircraft.
This can be either a stand at a gate or a remote parking stand.
The correct use of the word bay is for a holding bay as defined in ICAO ANNEX 14/I Aerodromes as:
Holding bay. A defined area where aircraft can be held, or bypassed, to facilitate efficient surface movement of aircraft.
So you use the term stand for an area where aircraft park (i.e. shut down engines, load/unload cargo, (dis-)embark passengers, etc.) and you use the term bay for an area where an aircraft temporarily holds (typically with engines running).
There are some sources using the term bay for parking, but I would consider all of them more informal than the official ICAO definitions:
The ENAV (company providing air traffic control services in Italy) writes (emphasis mine):
[...] which guides it right up to the parking bay.
on the English version of their website, but this could also be a wrong translation from the Italian original.
The Thales Group (French aerospace company) writes (emphasis mine):
Tower ensures smooth and seamless aircraft control between TMA boundary and the parking bay.
which again might be result of incorrect terminology used by the translator.
This article by Qantas suggests that the term is actually used by Australian ATCs (emphasis mine):
PILOT: “Qantas 635...we will arrive on blocks at 1500. Have 2 UMs, 2 WCHRs, 1 will need assistance to a taxi. Request a parking bay please.”
MOCO: “Copy that Qantas 635. 2 UMs, 2 WCHRs. You are for parking bay 26...2...6. See you on the ground Qantas MEL.
The CAAS (Civil Aviation Authority of Singapore) is also using the term (emphasis mine):
the aircraft was also assigned the shortest possible route from the runway to the parking bay to minimise fuel consumption.