The correct word is indeed THAT. The most familiar kind of relative clause after a human noun phrase modifies either the grammatical subject ("He's the coward WHO smeared my name in the press ten years ago") or a direct/indirect object ("He's not the coward [WHO/WHOM/THAT] I despised ten years ago.") Apple's multiple-choice sentence, however, is different.
This relative clause, "that he was," involves a subject complement. The subject complement and the grammatical subject of a sentence have identical reference ("He was a coward.").
Relative clauses with subject complements behave differently from other kinds of relative clauses. For one thing, you don't have to use a complementizer; it can be omitted. You can say "He is not the coward he was ten years ago."
If you do use a complementizer it's THAT, not WHO or WHOM*, (see note below). The complementizer in this kind of relative clause is not WHO, because it refers to an attribute of the person, not to the person him/herself. (We can see the same principle at work in the use of IT in a sentence such as "She was rich/a rich woman and she looked IT."**)
The subject complement in this kind of relative clause ("[the (X)] THAT [the person] is, was, etc.") characterizes that person: coward, weakling, monster, traitor, giant, genius, star, millionaire, benefactor, uncaring father, clever student. Or it carries the idea of "kind of [person, father, boss, neighbor, athlete]."
(Compare: "Fool THAT he was, he failed to see the dangers inherent in lending her seed money to open a goldfish spa." You also use the non-personal pronoun WHAT, not WHO, in a sentence such as
"” "I'll tell you WHAT (not *WHO) he was when we were in school. He was a dirty coward.")
Here is a sampling of examples from Google:
"” And maybe she's content to just be a top 15 player. She certainly isn't the fighter she was a couple of years ago.
"” Fortunately for Favre and Freeman, they know each other's tendencies inside out. But Freeman isn't the threat he was during his heyday in Green Bay
"” Dear "H,". It sounds to me like this young man is not the friend he was in the beginning of your friendship, for some reason.
"” Well my uncle had heard the story and being the coward that he was, and still is, he had made it a point to leave the home of the young woman he was courting
"” While peril lasted, Diane could still see her lover in the young Count; but out of danger, she despised him for the weakling that he was .
"” He's the same decaying garbage heap that he was ten years ago -- only more moronic, more mindless, more murderous
"” Many of his contemporaries looked on Snorri as the leader that he was and they were eager to have him as a friend.
"” It was his ability to communicate that made him the leader that he was.
"” Third, I want this man to be recognized for the Hero that he was and is today.
We were all heroes in those days (We Were a Band of Brothers).
* WHOM is ruled out because the personal pronoun does not represent any kind of object.
**The example is from Quirk et al., A Comprehensive Grammar of the English Language (Longman, 1985), p.349