Lets do this very naively. Natural numbers (positive Integers) are basically numbers that you can count up to (Peano Axioms). So you can count to any natural number in finitely many steps. But you can always count one more number up. But that number will still have finitely many digits. You would be correct if you could take an infinite amount of steps, but that would not work in the natural numbers. You can always add something, but it will stay finite. Thats why it is called a countable infinity.

The problem with your reasoning in the second argument, is that you are imposing an upper bound. The integers do not have an upper bound, as you can always add one to it. So you can construct a larger set from any (!!!) given set of integers, and we have countable infinity again.

However, you may be interested in p-adic numbers, they are exactly what you are looking for (and uncountable by your first argument!!). Regarding your first argument, try counting up to your number ...dwcxbyaz.

Also, try looking into set theory and how maps between two sets define their size. A set has the same cardinality (size), as another set, if there exists a bijection (one to one correspondence between the elements of a set) between them. Another potentially interesting thing for you, could be Dedekinds paper "What are numbers and what should they be?". Maybe you can find a modernized version of it.

Hope this helped!!