You are correct. Counting here changes its meaning.
The correct form should be "all the numbers reachable by counting in finite time".
E.g. you can count up to 100 in a minute. Up to a million in a month[1]. Up to a billion in quite a long, but finite time. You take all such numbers, and you say that a number named Omega comes just after all of them (just like million and one comes just after all numbers that are less than or equal to ine million).
Omega is your first infinte ordinal - or, simply, infinity.
And now you're counting in a new way.