CT Phipps
Carbon Dated and Proud
Rome was sacked in 410 by a band of Goths, not 376. That sack (and the wars with the Goths from 376 onwards) is indeed conventionally used as a point to mark the beginning of terminal decline (though most historians set the beginning of decline more generally earlier) since it was the first time a foreign army captured the city since the Gauls had in the 4th century BC (an event which arguably set in motion the building of the Roman empire as the Romans sought to expand the frontiers of their republic to ensure nothing like that ever happened again). It would be sacked several times thereafter, finally in 476.
The term "the Fall of Rome" refers most generally as you say to Odoacer deposing the last Western Roman Emperor (not at that time based in Rome I don't think), and the Roman senate (still based in Rome) transferring Imperial authority to the Eastern emperor, who in turn conferred status to Odoacer and the Goths as rulers of Italy.
Random aside but also the "Fall of Rome" was something made up by Enlightenment thinkers like Gibbon. It was political history (arguably all is) designed to demonize Christianity and exalt Roman "values" like stoicism and militarism and indicate that abandoning the latter led to the barbarians take over.
When Rome didn't "fall" so much as move and it only is a Fall if you consider the Eastern Roman Empire not to essentially exist (which Gibbon didn't because it wasn't Western European).
Which is to say that Gibbon was full of bullshit and agenda.