"Countably infinite" is a contradiction in terms. If it's countable, it's not infinite. By definition.
That reals are different than integers does not make either of them countable, or one of a "greater" infinity than the other.
Infinity is not a number. Thus, no arithmetic comparators apply to it. Terms like "greater" or "lesser," "more" or "fewer" are meaningless.
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.
A set is countably infinite if and only if there is a one-to-one mapping from the positive integers to the set in question. That's a definition. It's no more a "contradiction in terms" than any of the hundreds and hundreds of other definitions in math.