I think a bit of ground information is needed to understand which file to use.
Microsoft have two different applications for syncing with OneDrive.
The first (Which is the old one) used to be called SkyDrive and uses the GROOVE.EXE file. The Display name of this is called OneDrive for Business.
This application is just plain horrible and has caused Microsoft a ton of grievance over the years, its cache would get corrupt constantly and require a totalt resync of everything, it would mis-sync and was generally not usable for a work environment. I have seen tons of changes go lost and not be synced properly to OneDrive/Sharepoint from faults, errors and bugs with this application.
For the none Office 365 accounts they used an application simply called OneDrive with the OneDrive.exe as filename.
They then finally made it so that OneDrive.exe could sync with Office 365 (both OneDrive and SharePoint) and deprecated the old GROOVE.EXE.
If you download Office 365 from a new click to run installer it should no longer include OneDrive for Business (GROOVE.EXE).
If your users still use Groove.exe I would recommend that you upgrade them, for a much nicer and supported experience.
Microsoft has a nice article regarding deployment of the OneDrive.exe.
For searching properly in Google for more help be sure to search for either onedrive.exe or "Next Generation Sync Client"
