I found this blog by wondering the same question. Both sides have some good points.

For me though (so far), the ones who immigrated here are the "original" generation, and the first ones born here are the first generation. They were "generated" here.

I was born in Canada. I was conceived, born, and raised here.
My dad's parents moved here from England, and dad was born and raised here in Canada.
My mum's parents: sired and raised my mother in Scotland. My mother moved here when she was an adult and married my father here...

Uh, oh!!

So, paternally I'm second generation English Canadian. Maternally, I'm first generation Scottish Canadian! (or if you look at it the other way, for the debate here....etc. etc...). I didn't mean to make it more confusing, I just discovered this now!

I guess it comes down to a matter of one's own perception as to what generation they are. Perhaps what's best (and what's been mentioned before) is saying "First generation born here".