"Open Day" meaning that one does not have to go through an agency or other pre-screening to attend, and does not need a specific invitation (unlike an AD)
Emirates is not a Canadian/US airline therefore it is not wise to assume their definition of "open day" is the same as a Candian/US airline. I think you may be getting confused with the commonly used term "open house" which is as you describe- a 'drop in' at any time.
Emirates have chosen to conduct their recruitment in a certain way and have been doing so for years. Why would they suddenly change it for the US/Canada just because it's more convenient for some people that way? Their process takes a long time and if they had people turning up at 7pm it would go until the following lunch time!
As for attending an OD within six months- it had been mentioned recently by some that the rules have changed, where it used to be you could go to one OD after another so long as you did not attend an AD, now it seems they count sic months between each attempt. Perhaps it's to be more fair and give a chance to those who attend the first time and can't get noticed with all the repeat applicants who need more time to develop.
Since she took your CV (and saw you were late) I would guess the 6 months applies. Check the EK website to be sure and see what it says there.