Indeed; the rulebook is confusing because there's a side bar on "unit types" which talks about unit shapes, but it has been clarified in other rulings that when cards, etc, refer to "unit types", they are referring to the actual different unit names. For instance, "Skeleton Archers" and "Reanimates" are different unit types, despite them both having Triangle bases.
I think a more accurate set of terms would be:
Base Type - All units that share a common base shape are of the same "base type"
Unit Type - All units that share a common name are of the same "unit type"