The confusion is coming from you thinking Office 365 runs in "The Cloud", only the backend runs in the Cloud, the application installs and runs locally. There is not a special Online Version it is the same as the one from the OneDrive Web interface which is a simplified version that allows the most simple of editing.
Now you've confused me. If editing isn't the same I'm going with different versions in practice.No, Office Online is Office Online across the board. It is identical to buying the shrink-wrapped version except the licensing is different as it is purchased as a subscription, stop the subscription and you lose your license.