Am I imagining this or did London rear door buses have a sign advising that 'passengers must not give the moving off signal'? I interpreted the moving off signal as being a double push, which many passengers these days give as a request to stop. As noted above, some vehicles now convert a single push for a stop request into four bells which almost sounds like a train despatch communication.
To my recollection signs on bells (or more commonly in Edinburgh's centre door vehicles, a rubber strip in the ceiling) said 'Push Once'. Many children and a few adults would deliberately push twice as an act of defiance.
On rural single door vehicles around Edinburgh from the 1970s (Eastern Scottish, in common with Northern, Western & Highland as mentioned earlier) passengers were discouraged from touching the bell, whether proper front, half cab front or rear door, and especially if a conductor was on board. Having been told off for using the bell on one occasion in the early 1980s I didn't ring again on a single door vehicle until around 2004 and was almost never over-carried. Changes prompting universal passenger bell-pushing here seemed to be the advent of DDA plus main Edinburgh operator (Lothian Buses) phasing out their centre door fleet on which passengers always had to ring the bell to alight otherwise the driver wouldn't see them moving towards the door. Given the long time when both bus types were in operation with the same company nearly all passengers forgot the convention, and if sat upstairs you wouldn't be aware of the difference.
For relevance of DDA, see also the thread about whether passengers get out of their seat prior to the the bus pulling up at their stop. In bygone days people with mobility issues, prams etc weren't really allowed on buses so no worries with expecting passengers to move towards the door in advance of alighting. While there's usually a siren to help the most severely disabled passengers, a few others are likely to appreciate extra time for alighting and may not be able to shout to the driver from their chosen seat.
You'll get the occasional campanology enthusiast, either showing off or panicking, plus those who didn't hear or believe the first request. But most people realise only a single push is adequate and it's seldom that passengers in Edinburgh continually ring the bell in advance of their desired stop.