Are these correct?
More or less correct. Here's my take on the situation:
Transit passengers do not necessarily need to be on the same aircraft inbound and outbound, however they usually are. All that matters is that they're on the same 'flight' - which usually means the same flight number. The journey as treated as a single sector.
Transfer passengers arrive on one flight and leave on another (totally separate) flight. The journey is treated as two (or more) sectors.
For example, traveling LHR-Australia:
Transit flight:
BA15 LHR-(SIN)-SYD. You are ticketed LHR-SYD, it is effectively one flight. Bags are checked to SYD etc etc. During the transit period you have to stay within the transit area.
Transfer flight:
BA15 LHR-SIN
QF10 SIN-MEL. You receive two boarding passes for two flights. The ticket as treated as two sectors. Bags may be checked to SIN (for collection and subsequent drop-off), or all the way through to MEL. Depending on time between your flights you are free to do what you like whilst you are in SIN.