Based on what you’ve described, you would be applying under Paragraph A57C of the Immigration Rules. rather than Appendix V.
Requirements for entry clearance or leave to enter – Short-Term Student
A57C. All applicants for entry clearance or leave to enter the UK as a short-term student must meet the following requirements:
(a) the applicant does not fall for refusal under the general grounds for refusal; and,
(b) meets all of the following requirements. The student:
(i) is aged 18 or over.
(ii) does not intend to study at a state-maintained school or institution.
(iii) does not intend to study in the UK for extended periods through frequent or successive periods as a short-term student.
(iv) does not intend to take employment, including paid or unpaid work, a work placement or work experience in the UK.
(v) does not intend to undertake self-employment or engage in business activities or any professional activity in the UK.
(vi) has enough funds to meet the cost of his return or onward journey from the UK.
(vii) will be maintained and accommodated adequately out of funds available to him.
(viii) will not have recourse to public funds
(ix) is genuinely seeking entry as a short-term student.
You ask a fair question:
[W]hich one is correct, six calendar months (184 days, in my case), or 180 days?
Indeed, numerous attempts have been made at normalizing the day counting methods and defining what 6 months is in terms of days. All of these have failed. The day counting issue is ambiguous by intent, and it could be unwise to appear at the border and have that debate with an Immigration Officer. The number of days are not relevant. When an Immigration Officer considers an application vis-a-vis the course description, they will know what to do next.
While visa-free nationals may apply for A57C at the border, you would want to double check with your school’s Authorising Officer before doing so.