Reality Check - SharePoint Online, Do you need to be a Term Store Administrator to manage Terms?

In SharePoint Online I just added two people as Term Store Administrators so they could manage Terms and Term Sets in a Term Group. I had already added one of them as a Term Group Manager and the Other as a Term Group Contributor to the Term Group, and I thought that would be enough to allow them to add\create\edit Terms in the Term Group.  I didn't think I needed to add them as Term Store Administrators.   My current understanding is that they must be added as a Term Store Administrator and either a Term Group Manager or Term Group Contributor to manage terms in a term group.  I guess this is delegated management because you have to be either an O365 Global Admin or SharePoint Admin to add people as Term Store Administrator via the O365 Admin Center. It also appears that Term Store Administrators can only manage Terms in Term Groups on which they are a Term Group Manager, It also seems like you would have to be a Term Store Administrator to add manage Terms in a Term Group as a Term Group Contributor.    Would someone set me straight if I have some part of this wrong? Thanks - Greg