The word "am" is a symmetric verb in this case, it functions much like an equals sign. One one side is "I" and on the other side is the phrase "the one who wants to be with you". So the fact that "I" is first person all the way on the other side of the 'equation' doesn't affect "wants".
So, looking at the noun phrase "the one who wants to be with you", who or what "wants"? It is "the one" who wants. This is third person singular, so the form should be the same as it is in "he wants" or "it wants".
The traditional pedantic version of this sentence would be "It was he whom I voted for" (although a pedant who believed in avoiding stranded prepositions would instead say "It was he for whom I voted").
The object of the preposition for in the relative clause is typically analyzed as consisting only of the relative pronoun whom/who. The relative pronoun and the personal pronoun he/him refer to the same person, but they do not occupy the same position in the grammar of the sentence. The personal pronoun he/him is not part of the relative clause at all.
The "Let he/him who..." question has "he/him" as the object of the main clause and "who" as the subject of the relative clause. It seems comparable to your sentence, where "he/him" serves as a predicative complement in the main clause and "who/whom" serves as the object of the preposition "for" in the relative clause. In both cases, the standard prescriptivist viewpoint is that the relative pronoun should be inflected according to its role in the relative clause, and the pronoun before the relative pronoun should be inflected according to its role in the main clause.
"I chose whoever came first" is a different situation because there is no other pronoun before the relative pronoun: it is a "fused" relative. A fused relative pronoun is "supposed" to inflect according to its role in the relative clause. See this blog post: "For Whomever the Bell Tolls" (by Jonathon Owen, Arrant Pedantry). The actual, as opposed to prescribed, behavior of "fused relatives" seems to be fairly complicated: see F.E.'s answer to “Put me in touch with whomever created it”? for more details.
The verb be is not actually analyzed as taking a direct object, but a "predicative complement". Completely separately* from the issue of relative clauses, there is variation between "he" and "him" as a predicative complement: "It was him" is usual, but "it was he" has traditionally often been prescribed as a "more grammatical" form (based on the idea that the predicative complement should have the same case as the corresponding subject, which in this case is the nominative pronoun it). Sentences with nominative predicate pronouns, like "It was he", still exist to some extent as "elevated" variants of sentences with accusative predicate pronouns.
(*"Completely separately" may be a slight simplification: Barrie England's answer to "It is I who am at fault?" indicates that the use of "nominative" predicative complements may in fact be more frequent in present-day English before relative clauses that have who as the subject. But from a prescriptive point of view, there isn't supposed to be any relationship between the use of "It was he" and the presence of or case of a following relative pronoun.)
Best Answer
Generally speaking, in English, accusative (also known as “objective”) pronouns (like me) are the “default” form. That is, unless there is a specific syntactic rule requiring use of a different case, such as nominative (I), genitive (my/mine), or reflexive/intensive (myself), in English you use the accusative case.
In the syntactic context where a pronoun is not serving a role relative to an explicit verb, such as when it is the simple answer to a question, or if one is labeling something, such as a photo, accusative pronouns are standard. “Who wants to come?” “Me.” Nominative pronouns are impossible here—you cannot answer the question “Who wants to come?” with “I”, nor would anyone label a photo “I”.
This holds even if negated: “Who wants ice cream?” “Not me.”
If you want to use the highest register, most formal English, however, you should avoid the question of what case to use with pronouns standing alone, and use a complete sentence: “I do not want ice cream.”