I don't remember seeing that particular Help page before - thanks for the link. And now I feel quite confused about the rel-canonical recommendation on that page.
can the URL in the alternative be on a different website
It isn't explicitly stated, is it? I'd guess that if a different subdomain is OK then a different domain altogether should be, too - but after reading that rel-canonical comment I'm not too confident of my guesswork right now.
And rereading this Help section, I see it is about "any website that offers content in more than one language." Maybe they feel they already have the ccTLD situation under control.