Such a division seems to cause more harm than good, for several reasons. First, recognizing only two categories is not representative of the variety of uses of inheritance, reflected by the preceding classification. Since no one will advocate introducing ten different language mechanisms, the result will be too limitative. One can find arguments to support either answer. It is also interesting to think of the consequences that such a division will have on the complexity of the language.
Inheritance comes with a number of auxiliary mechanisms.
Renaming is definitely useful for module inheritance. In the modeled external system, variants of a certain notion may introduce their own special terminology, which it is often desirable for the software to respect. Repeated inheritance may occur with either form. As always when we introduce new mechanisms into a language, they interact with the rest, and with each other. All this for the benefit of a purist's view of what inheritance should be.