The third designation is Ruh al-Qudus (Holy Spirit), described here within a philosophical-theological framework involving nonmaterial or intellectual realities. In that setting “spirit” is not automatically limited to the ordinary notion of a soul animating a human body.
Two questions are separate: what place the Holy Spirit holds in the doctrine under discussion, and how the Qur’an employs the expressions “spirit” and “Holy Spirit” in particular contexts. The appearance of a common term does not establish identical referents or doctrine.
The methodological distinction is shared wording versus shared meaning. Identical labels can function differently in distinct theological systems; serious comparison examines the actual meaning, argument and setting of each use.