You are mistaken. There are infinitely many integers, because, for any integer, one can always obtain a larger one. However, they are countable in that they can be ordered, giving each one a place, with a definite “next one” and a definite “previous one” - in the case of every single integer.
In particular, for any given integer, there is always a “next one” and so the set of integers is infinite. (Indeed, even the set of “positive integers” is infinite.)
Just because something is countable doesn’t mean it isn’t infinite. In order to be not infinite, there has to be a finite number of them - a last one of them. The number of ways a deck of cards can be ordered is finite. It’s a huge number, but, it is a definite number, and there are no other ways the deck can be ordered other than as one of those ways. But there is no last integer; one can always get a larger integer by taking the factorial, by squaring it, or even, modestly, by just adding 1.
I think I understand the distinction. It seems a little bit like semantic hair-splitting, but I’ll concede the point.
In fact, despite the fact that poker, solitaire (and other card games) have been played for hundreds of years, it is highly likely that a properly shuffled deck has ever been in the same order.
Put another way, every shuffled 52-card deck has had its own unique order which has likely never been duplicated.
I find that to be an amazing fact and yet it doesn't even come close to describing infinity.