Not quite.
Both a short circuit and overload may trip a breaker through the same mechanism, but an overload is typically a MUCH smaller amount. A short circuit can easily generate several hundred, thousand, or even tens of thousands of amps for a short period of time. This is where the breaker's AIR rating comes in - completely different than it's thermal protection rating. The breaker needs to be able to quickly interrupt these huge loads before it gets welded together and all the wiring along the way vaporizes

An overload heats up the breaker's thermal trip slowly, and the breaker will handle small overloads over time.
Ground faults are rather completely different. Many/most ground faults will not trip a typical breaker at all. Your smallest breaker is usually a 15A... based on several factors it will actually trip at different currents, but not below <15A.
A GFI can easily pick up a 4ma current leak (ground fault) and trip. It would take over 3700 times this current at a MINIMUM to trip the breaker (actually far more, again depending on the breaker's actual trip curve) under what a normal breaker is designed to do - over current and short circuit protection. And it would take some time to do this! A typical circuit breaker has no idea that the current is not traveling back on the intended path, so unless the ground fault exceeds its thermal overload rating it will not trip and it will think its a completely normal load. GFIs also have a bit of a different switching mechanism in them than a circuit breaker.