It is definitely not "definitely the other way around".
POST is meant for anything and everything, absolutely including updates. PUT is meant for writing-or-overwriting a complete or partial resource, at a URI that is the Request URI. In practice, for most applications, this makes PUT impractical for new entities (because the requester cannot know the correct URI for the new resource), but in some applications it can make sense, and either way there's more to it than you assert.
POST is meant for anything and everything, absolutely including updates. PUT is meant for writing-or-overwriting a complete or partial resource, at a URI that is the Request URI. In practice, for most applications, this makes PUT impractical for new entities (because the requester cannot know the correct URI for the new resource), but in some applications it can make sense, and either way there's more to it than you assert.