Since the Pierce-Karloff-Whale Frankie is as canonical as Shelley’s original, there are two answers—No and Yes.

The PKW Frankie is not a proper zombie

1. No organic unity — In fact, he is not even a proper “he,” having as he does only the false unity of a clump, quilt, or assemblage. Many dead body parts from different donors are combined together. What unifies them into “a person” is the same principle that unifies unassembled humans—i.e., the fact that all parts are coordinated by one and the same executive power.
2. Ordinary metabolism — What reanimates the PKW Frankie is … ordinary mechanical force. What sets things moving in the life-way is … just moving them. This is done with electricity, but the principle is the same—ordinary mechanical engineering. The PKW Frankie is alive because life is just material motion, and Frankie’s parts are (1) properly arranged and (2) properly kickstarted.

So aside from the multiple donors-issue, Frankie is just an ordinary person, and his animation, while artificial, is not unnatural. The power that animates Frankie is the same power that animates you and me. And “having been previously dead” is no big deal. The life in Frankie post-storm is no stranger than the life in Granny post-defibrillator.

The Shelley Frankie may be a proper zombie

everything

has

animation

The (corporeal) undead

— do WIKI part

— then say, "As usual, when it comes time to be fully clear and distinct about the Western cultural legacy, we must turn to Gygax and his noble descendants, who have been tasked by our desire for realistic interaction to unpack such vague entities into maximal particularity required for dice determination, computer simulation, and ultimately AI emulation.

From Monster Manual (5e)

Undead are once-living creatures brought to a horrifying state of undeath through the practice of necromantic magic or some unholy curse. Undead include walking corpses, such as vampires and zombies, as well as bodiless spirits, such as ghosts and specters.

Intentional reanimation

— by a demon

— by a cleric

— Beholders and Death Tyrants can transform former slaves and enemies into undead servants.