If the first chromosomes in the parent cell have two chromatids, and these split, then wouldn't the resulting cells be haploid with each having one chromatid turned into chromosome, or only one copy of each chromosome?

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The resulting cells would not be haploid unless they were undergoing meiosis II. Because the cells undergo DNA synthesis during the S-phase of interphase, then the cells get enough chromatids so that when they split apart, there is an equal amount of genetic information in each cell as a result. The cells are still diploid, they just make copies of that same DNA before splitting, making it so they only have one set of genetic information, just two copies of it that can go to different cells during mitosis. I hope that helps.