IMHO, I find "Module's homepage" better than "Online documentation". I'd actually prefer "Project page" over the first two.
The reason I think online documentation is not the best way to word it is because it does not link to the online documentation. Clicking on "online documentation" and landing on project page would be annoying (for me, at least). If we say "online documentation", we should link to the documentation that's available for some modules (accessible from sidebar on project page). Of course, we can't get that link without an API call, and I don't want the scope of this issue to expand to that.
I understand the reasoning in #11 but I'd point out that in the help text where it is written as "online documentation", it actually points to the documentation. Also, there is no such text for contrib modules help entries out of the box.