I think it is too soon to make any demands of the supervisor to answer by a certain date or anything like that, nor is there a need to ask him to explain whether you heard him correctly, or to confirm or deny anything. All you need to do at this point is to lay out your understanding of events - if the supervisor wants to contest that understanding, he/she is at liberty to try to do so. A key principle for this sort of email is that you should be very clear and factual and maintain a neutral and polite tone. Something like the following:
Dear [name of supervisor],
I am writing in connection with our recent meeting on [insert date] to discuss my status and funding in the program. As you know, my acceptance letter to the program, dated [insert date of acceptance letter (you have a copy right? maybe attach it to the email)], states that I am entitled to six years of funding. No conditions were attached to this promise other than my maintaining my status as a PhD student in good standing in the program, which has been and remains the case. [obviously, only write all of that if those are true statements...]
During the meeting, you stated however that I am not entitled to a sixth year of funding, based on your assertion that "We spoke previously. I asked whether you needed funding this year and you shook your head." [Note: if you put their words in quotes, make sure it is an accurate quote.] I would like to clarify that I never indicated that I did not need funding for a sixth year, either by a shake of the head or in any other way. I do in fact need and want funding, and was and still am expecting to receive it based on the department's promise, signed by [insert title: Graduate Program Chair/Department Chair, etc] [insert name of signatory to the letter] in the acceptance letter I mentioned above.
I would appreciate if you reply to this email at your earliest convenience to acknowledge that you received it. Thank you for your continued support and for your assistance in correcting any misunderstanding that may have occurred.
Once the supervisor replies, if they explicitly confirm that you will get the funding, great; if they do not explicitly confirm that but don't contest your understanding, you have now put yourself in a good position to go to the department head; and if they try to contest your description (like claiming that you did shake your head), I think they will appear rather foolish for relying on such signals rather than statements documented in writing or even a verbal statement, so again you are in a good position to go argue your case with the department head.
Note also that I added a polite request for an acknowledgement and nothing more. The supervisor is free to ignore that, but again doing that will make them look bad, and in any case if you don't get a reply within a few days, you can and should proceed to the next stage of the plan.
One other option I can think is to copy the department head on the email. That is a bit of a "nuclear option" and is likely to be interpreted as an aggressive and antagonistic move (nobody likes to be made to look bad in front of their supervisor), so probably it's too soon for that level of response. But it's something to consider in the event that you want to dial up the level of forcefulness of your response.
Hope this helps, and good luck!