Hi Melanie,
> I found a lot of documentation online, but things are really not clear to me.
To help clarify, there are 2 different options to enable versioning - where you enable multi-user editing of the same data in an enterprise geodatabase: traditional versioning and branch versioning. Some resources:
- Versioning 101 - ArcUser Technical Article (traditional versioning, this is an older article, but still relevant)
- Branch Versioning—What is it and How Does it Work?
> Do I need a branch version to edit my feature layers in portal then reconcile and reinject
You would enable versioning at the geodatabase tier (via ArcGIS Pro or ArcMap). You can choose one of the versioning models above, see this blog to help you decide which To Branch or Not to Branch
> If I checked "register the objects with the option to move edits to base", do I need a branch version?
The "register the objects with the option to move edits to base" is applicable to traditional versioning,
Versioning types - help topic
> In ArcGIS Pro I cannot see the option "Version Management".
Are you unsure where to locate it in ArcGIS Pro?
Manage versions—Geodatabases | Documentation
Hope this helps,