Says the various public health officials and organizations

I repeat: who else? You saying so doesn't count, 'cause who are you?

What vaccines do is generate immunity, and stop infections because the immune system attacks the(in this case of viral vaccines), before the host can reach a point that they become symptomatic and spread the disease through standard vectors.

Being vaccinated does not mean you cannot be infected and spread the infection, but statistically you are less likely to get seriously sick or die, and in some cases - depending on which vaccine and which variant - you may pass the virus on at a much reduced rate.

In Canada, like the US - unless you die, they don't count your admission.

Pure, unadulterated nonesense. You aren't counted unless you test positive.