My first impression was that the source you've quoted is contradictory nonsense.
Having spent a bit of time reading up, I've come to the following conclusion.
The notion of an "abstract data type" makes sense only to the extent that we recognise certain data types can have different but equivalent implementations.
What might be abstract about an "integer" is that part of the type that doesn't seem to depend on whether a machine is big-endian or little-endian, two's complement or one's complement, and so on.
It's worth bearing in mind that <PERSON>, who apparently coined this term, was writing in 1974, when these sorts of storage details loomed large.
However, what exactly the full "abstract" definition of any given type is, may be something that is itself open to different interpretations. It's not always clear that programmers are dealing in rigorous abstractions, or merely in imprecise definitions and sloppy thinking.
I would be inclined to say that an "abstract" integer, free of all storage details like field size, is not something that is widely known to programmers. Probably because integers with variable-size storage (and thus, bounds limited only by the total available storage, which is a kind of limitation that applies only to the machine, not to a data type) don't commonly exist, and are probably extremely slow to process.
So we programmers might deal with the abstract Int32, and the abstract UInt8, but not the abstract integer that has no fixed bounds.
It's also not clear that "fixed-width integers", as a general concept encompassing a family of fixed-width integer types, meets the criteria of being an ADT in its own right, because each different fixed-width integer type do not differ merely by their implementation details, but by having a different fundamental specification of the range of values that can be represented by each fixed-width type.
I would also say the general usefulness of the distinction between "abstract" and "implemented" has probably reduced since 1974, as the use of compilers means that probably all types are conceived of by the programmer in a way that is abstract from the minutiae of their binary implementation. 904