Since there are options “Associated partners” and “Partners” in the list of participants in part A of the proposal, should the applicant include associated partners in their proposal?

Yes, as indicated in the GfA (page 9) https://rea.ec.europa.eu/system/files/2021-10/MSCA%20COFUND%202021%20-%20Guide%20for%20Applicants.docx.pdf: Associated partners known at the proposal stage must be included under the participants section in the part A of the proposal as well as in the relevant section in the part B2 (section 5).