Word Online is great for general editing, and its great for real-time collaboration, but it doesnt have advanced formatting features that you might need for more sophisticated document composition. If you have to choose between Word 2013 and Word Online for co-authoring, you might lean toward Word Online if you can forego fancy stuff like fiddling with the TOC and advanced table layouts.
Word 2016 has the ideal tools for co-authoring, with the share button in the upper right, the share panel giving easy access to co-authors and real-time editing, all in the full-featured version of the software.